1. Krzysztof Barański. slides. (2022).

In the Token sliding problem, one starts with a graph and independent sets I_s, I_t. We put k tokens on vertices of I_s and ask whether it's possible to reach I_t after a finite sequence of moves, where 1 move is sliding 1 token along the edge so that no 2 tokens are adjacent at any point. It was shown in 2021 that this problem is W[1]-hard for graphs of girth 4 or less. In this presentation, we will see how the problem becomes Fixed-parameter tractable for the other graphs (girth 5 or more).

1. Valentin Bartier, Nicolas Bousquet, Jihad Hanna, Amer E. Mouawad, Sebastian Siebertz. Token sliding on graphs of girth five. arXiv:2205.01009. (2022).
2. Demian Banakh. Token sliding on graphs of girth five. slides. (2022).

A standard Turán problem for a graph F is to determine the maximal number of edges in a graph not containing F as a subgraph. This problem for directed cycles in oriented graphs is trivial, but its various generalizations, when one asks for minimum outdegree or number of other subgraphs, occurred to be hard problems. In particular, finding minimum outdegree (or semidegree) forcing an oriented graph to contain a directed triangle is a Caccetta-Häggkvist conjecture, which is open for 45 years despite numerous partial results. During the talk we will present a solution (obtained with Jan Volec) to a conjecture of Kelly, Kühn and Osthus on the minimum semidegree forcing an oriented graph to contain a directed cycle of any given length at least four. We will also discuss results (obtained jointly with Justyna Jaworska, Bartłomiej Kielak and Tomasz Ślusarczyk) for the generalized Turán problem for directed cycles when one maximizes the number of directed cycles of some other length.

1. Rudolf Fleischer, Jiajin Yu. A Survey of the Game “Lights Out!” In: Brodnik, A., López-Ortiz, A., Raman, V., Viola, A. (eds) Space-Efficient Data Structures, Streams, and Algorithms. Lecture Notes in Computer Science, vol 8066. Springer, Berlin, Heidelberg. (2013).
2. Bartłomiej Błoniarz. slides. (2022).

1. Gregory Gutin. Note on Perfect Forests. arXiv:1501.01079. (2015).
2. Jakub Dziarkowski. slides. (2022).

Treewidth is a fundamental graph parameter that, informally, characterizes how tree-like a graph is. We give a 2^{O(k^2)}·n^O(1) time algorithm for determining if the treewidth of a given n-vertex graph is at most k and outputting the corresponding tree decomposition. This resolves the long-standing open problem of whether there is a 2^{o(k^3)}·n^O(1) time algorithm for treewidth. In particular, this is the first improvement on the dependency on k in fixed-parameter algorithms for treewidth since the 2^{O(k^3)}·n^O(1) time algorithm given in 1991 by Bodlaender and Kloks, and independently, by Lagergren and Arnborg. We also give a k^O(k/ε)·n^O(1) time (1+ε)-approximation algorithm for treewidth.

Joint work with Daniel Lokshtanov.

Fixed point combinators (and their generalization: looping combinators) are classic notions belonging to the heart of λ-calculus and logic. We start with an exploration of the structure of fixed point combinators (fpc’s), vastly generalizing the well-known fact that if Y is an fpc, Y (SI) is again an fpc, generating the B ̈ohm sequence of fpc’s. Using the infinitary λ-calculus we devise infinitely many other generation schemes for fpc’s. In this way we find schemes and building blocks to construct new fpc’s in a modular way. Having created a plethora of new fixed point combinators, the task is to prove that they are indeed new. That is, we have to prove their β-inconvertibility. Known techniques via B ̈ohm Trees do not apply, because all fpc’s have the same Bohm Tree (BT). Therefore, we employ ‘clocked BT’s’, with annotations that convey information of the tempo in which the data in the BT are produced. BT’s are thus enriched with an intrinsic clock behaviour, leading to a refined discrimination method for λ-terms. The corresponding equality is strictly intermediate between =β and =BT, the equality in the classical models of λ-calculus. An analogous approach pertains to L ́evy–Longo and Berarducci trees. Finally, we increase the discrimination power by a precision of the clock notion that we call ‘atomic clock’.

The theory of sage birds (technically called fixed point combinators) is a fascinating and basic part of combinatory logic; we have only scratched the surface.

The paper focuses on the representation problem in political elections, using a theorem from number theory. A. Katz's work gives an answer to the question - of whether there exists a way to construct the election list so that it does not matter how many politicians are selected and the politically different groups of the party will be represented?

1. Amoz Kats. Can a party represent its constituency? Public Choice. 44, 453-456. (1984).
2. Julia Biały. slides. (2022).

We consider internal partitions of graphs, which is a partition of V into two sets, such that every vertex has at least half of its neighbors in its own set. Several investigators have raised the conjecture that d-regular graphs always have an internal partition, assuming their set of vertices is big enough. Here we prove this conjecture for d=6. We also investigate the case when |V|=d+4, which leads to some new problems on cubic graphs, and find new families of graphs that don't have an internal partition.

1. Amir Ban, Nati Linial. Internal Partitions of Regular Graphs. arXiv:1307.5246. (2013).
2. Katzper Michno. slides. (2022).

In 1964 Vizing proved that to properly color the edges of a graph G, one need at most ∆+1 colors, where ∆ is the maximum degree of G. In his paper, Vizing actually proves that one can transform any proper edge coloring into a (∆+1)-edge-coloring using only Kempe changes. Soon after his paper, he asked the following question: is an optimal edge-coloring always reachable from any proper edge-coloring using only Kempe changes? Bonamy & al. proved that the conjecture holds for triangle free graphs, following their work, we prove that it holds for all graphs.

Gallai proved that the vertex set of any graph can be partitioned into two sets, each inducing a subgraph with all degrees even. We prove that every connected graph of even order has a vertex partition into sets inducing subgraphs with all degrees odd, and give bounds for the number of sets of this type required for vertex partitions and vertex covers. We also give results on the partitioning and covering problems for random graphs.

1. A.D. Scott. On Induced Subgraphs with All Degrees Odd. Graphs and Combinatorics. 17, 539-553. (2001).
2. Jakub Siuta. On Induced Subgraphs with All Degrees Odd. slides. (2023).

König's theorem lets us determine the maximum number of pairwise independent edges in a bipartite graph. In the paper, L. Lovász focuses on critical graphs, meaning that if any of their edges are removed, the size of maximum matching diminishes. Considering a certain generalization of the above-mentioned concept, Lovász gives a simple condition that is necessary and sufficient for a graph to be critical. The result is used to solve a conjecture by Erdős regarding strict hypergraph coloring.

1. L. Lovász. A generalization of Kónig's theorem. Acta Mathematica Academiae Scientiarum Hungaricae. 21, 443-446. (1970).
2. Aleksander Katan. A generalization of Konig's theorem. slides. (2023).

We will provide a gentle introduction to the on-going work on constructing a structural theory for graph classes defined by forbidding obstructions definable in logic. The focus will be on monadically stable classes of graphs: classes where one cannot define arbitrary long total orders using a fixed first-order formula. We will review recent advances on characterizing these classes in a purely combinatorial manner, in particular through a game model: the Flipper game.

In the extremal graph theory, there are many unsolved problems related to the finding of sparse subsets in graphs. The most famous one, stated by Erdos in 1976, asks whether every triangle-free graph contains n/2 vertices that span at most 1/50 n² edges. In our work we consider, and successfully prove, a modified version of this theorem which conjectures that every K₄-free graph has n/2 vertices spanning at most 1/18 n² edges. This bound is tight, as the balanced blow-up of a triangle is an extreme example. We achieve the proof by strengthening some of the previous results and by stating some new arguments which show that the only K₄-free graph which has at least 1/18 n² edges in every half is the blow-up of a triangle.

1. Christian Reiher. K₄-free graphs have sparse halves. arXiv:2108.07297. (2021).
2. Hubert Zięba. K₄-free graphs have sparse halves. slides. (2022).

Ando conjecture states that we can partition vertices of any cubic graph into two parts that induce isomorphic subgraphs. We show that this conjecture is true for sufficiently large connected cubic graphs. In the proof, we use probabilistic methods with recoloring arguments.

1. S. Das, A. Pokrovskiy, B. Sudakov. Isomorphic bisections of cubic graphs. Journal of Combinatorial Theory, Series B. 51, 465-481. (2021).
2. Łukasz Selwa. Isomorphic bisections of cubic graphs. slides. (2022).

Let us say that a graph of maximum degree Δ has local density at most η if the number of edges spanning any neighbourhood is at most η·(Δ choose 2), i.e. if the edge density is no more than an η fraction of the maximum possible. What is the largest chromatic number of such graphs? When η=0, this corresponds to asking about the largest chromatic number in triangle-free graphs of maximum degree Δ. This goes back to an old question of Vizing and is the objective of a recent breakthrough of Molloy. It is natural — and also connects to various other problems in the field — to consider other choices for η. We will broadly discuss this problem, including its classic origins in Ramsey theory, and some different ideas that have recently proven fruitful.

This will touch on recent joint works with Davies, Hurley, de Joannis de Verclos, Pirot, and Sereni.

The 1-2-3 conjecture asserts that for every connected simple graph of order at least 3 edges can be weighted with 1,2 and 3 so that each pair of adjacent has different weighted degrees. We consider a modified version of this conjecture with 1,2 weights only. By using f-factors modulo k of the graph, we prove it for non-bipartite (6𝛘(G)-5)-edge-connected graphs and completely characterize bipartite graphs having this property.

1. Carsten Thomassen, Yezhou Wu, Cun-Quan Zhang. The 3-flow conjecture, factors modulo k, and the 1-2-3-conjecture. Journal of Combinatorial Theory, Series B. 121, 308-325. (2016).
2. Hubert Zięba. The 3-flow conjecture, factors modulo k, and the 1-2-3-conjecture. slides. (2022).

Hadwiger's conjecture is an important conjecture in graph theory which states that every graph without a K_t-minor is (t-1)-colorable. This conjecture does not extend to list colorings, but Kawarabayashi and Mohar (2007) conjectured that there exists a constant c such that every graph with no K_t-minor has a list chromatic number at most c·t. More specifically, they conjectured that c = 3/2 is sufficient. Refuting the latter conjecture, we prove using the probabilistic method that there exist graphs with no K_t-minor with list chromatic number at least (2-o(1))·t, and hence c ≥ 2 is necessary. This improves the previous best-known lower bound by Barát, Joret, and Wood (2011), who proved that c ≥ 4/3.

1. Raphael Steiner. Improved lower bound for the list chromatic number of graphs with no K_t minor. arXiv:2110.09403. (2021).
2. Tomasz Mazur. Improved lower bound for the list chromatic number of graphs with no Kt minor. slides. (2022).

In 1954 Kővári, Sós, and Turán showed that every n-vertex graph not containing K_s,t has at most O(n^2−1/s) edges. We construct graphs matching this bound with t≈9^s, improving on factorial-type bounds. In this talk, I will explain probabilistic and geometric ideas behind the construction.

Constrained graph orientation problem deals with directing graph edges such that graph vertices fulfills some conditions. Here, we are focusing on contant indegree orientations of maximal planar and similar classes of graphs. We analyse the relationship between such orientations and other combinatorial properties of these graphs, including the existence of particular decompositions into trees given by the famous Nash William's theorem.

1. H. de Fraysseix, P. Ossona de Mendez. On topological aspects of orientations. Discrete Mathematics. 229(1-3), 57-72. (2001).
2. Grzegorz Gawryał. On topological aspects of orientations. slides. (2022).

It is known that the number of edges in a minimal non-2-colorable hypergraph is at least as high as the number of its vertices. We show the link between this and the fact that a minimal unsatisfiable CNF formula with n variables must contain at least n + 1 clauses. We show different proof of these facts and give infinite versions. We also analyze the structure of minimal unsatisfiable CNF formulas with exactly n variables and n + 1 clauses.

1. Ron Aharoni, Nathan Linial. Minimal non-two-colorable hypergraphs and minimal unsatisfiable formulas. Journal of Combinatorial Theory, Series A. 4(2), 196-204. (1986).
2. Rafał Kilar. Minimal Non-Two-Colorable Hypergraphs and Minimal Unsatisfiable Formulas. slides. (2022).

Whereas interior point methods provide polynomial-time linear programming algorithms, the running time bounds depend on bit-complexity or condition measures that can be unbounded in the problem dimension. This is in contrast with the simplex method that always admits an exponential bound. We introduce a new polynomial-time path-following interior point method where the number of iterations also admits a combinatorial upper bound O(2ⁿ n^1.5 log n) for an n-variable linear program in standard form. This complements previous work by Allamigeon, Benchimol, Gaubert, and Joswig (SIAGA 2018) that exhibited a family of instances where any path-following method must take exponentially many iterations.

The number of iterations of our algorithm is at most O(n^1.5 log n) times the number of segments of any piecewise linear curve in the wide neighbourhood of the central path. In particular, it matches the number of iterations of any path-following interior point method up to this polynomial factor. The overall exponential upper bound derives from studying the ‘max central path’, a piecewise-linear curve with the number of pieces bounded by the total length of 2n shadow vertex simplex paths.

This is joint work with Xavier Allamigeon (INRIA/Ecole Polytechnique), Daniel Dadush (CWI Amsterdam), Georg Loho (U Twente), and Bento Natura (LSE/Georgia Tech).

The Farey sequence F_n is the set of rational numbers a/b with 0 ≤ a ≤ b ≤ n and gcd(a,b) = 1. In 1970, Graham proposed the following conjecture. Let a₁, a₂, ..., a_n be distinct positive integers. There exist indices i ≠ j, such that we have a_i/gcd(a_i,a_j) ≥ n. In the paper, the authors show interesting properties of Farey sequence sets and how they are closely related to Graham's problems.

1. Liuquan Wang. Farey sequence and Graham's conjectures. arXiv:2005.04429. (2020).
2. Szymon Salabura. Farey sequence and Graham’s conjectures. slides. (2022).

A graph G is k-color-critical if G is not (k-1)-colorable, but every proper subgraph is. For S, an orientable surface other than the sphere, there are infinitely many k-color-critical graphs if and only if 2<k<6. For k>4 there is the polynomial algorithm for deciding if a graph can be colored with k colors. In this paper, the authors prove those theorems and show some results for list coloring.

1. Carsten Thomassen. Color-Critical Graphs on a Fixed Surface. Journal of Combinatorial Theory, Series B. 70(1), 67-100. (1997).
2. Katarzyna Kępińska. Color-Critical Graphs on a Fixed Surface. slides. (2022).

The essentially free variables of a term t in some lambda calculus $FV(t)$ form the set $\{x : \forall t =_{beta} u \rightarrow x\in FV(u) \}. This set is signicant once we consider equivalence classes of \lambda terms rather than \lambda terms themselves as for instance in higher order rewriting. An important problem for (generalised) higher order rewrite systems is the variable containment problem. This property is important when we want to consider $t \rightarrow u$ as a rewrite rule and keep n-step rewriting decidable. Variable containment is in general not implied by $FV(t) \supseteq FV(u)$. We give a decision procedure for the variable containment problem of the second order fragment of $\lambda^\rightarrow$. For full  $\lambda^\rightarrow$ we show the equivalence of variable containment to an open problem in the theory of PCF;  this equivalence also shows that the problem is decidable in the third order case.

A q-factor of a k-regular graph is its q-regular subgraph covering all vertices. q-factorization is a partition of edges of a graph into disjoint q-factors. For q-factorization to exist it is necessary that q\mid k. It was proven that for even q the converse is also true - qd-regular graph has a q-factorization. The paper investigates when qd-regular graph with odd q admits q-factorization, given additional assumptions like planarity and/or high connectivity.

1. Carsten Thomassen. Factorizing regular graphs. Journal of Combinatorial Theory, Series B. 141, 343-351. (2020).
2. Filip Konieczny. Factorizing regular graphs. slides. (2022).

The problem of determining a sequence of points on the unit circle is considered, such that at any time t the lengths of the segments (sticks) resulting from splitting the circle at the locations set by the first t points are as equal as possible. The authors consider the sequence x_k=lg(2k-1) mod 1 discovered and analyzed by De Brujin and Erdos in 1949 called the log stick-breaking strategy, proven to be optimal under 3 selected measures. The analysis of this sequence is extended by showing an interpretation in which log stick-breaking is a uniquely optimal strategy, and a more general framework is designed in which the optimality of this strategy can be explored.

1. Lyle Ramshaw. On the gap structure of sequences of points on a circle. Indagationes Mathematicae (Proceedings). 81(1), 527-541. (1978).
2. Hubert Dej. On the Gap Structure of Sequences of Points on a Circle. slides. (2022).

Praca opisuje "zderandomizowany" odpowiednik podnoszenia grafu do kwadratu. Nowa operacja zwiększa spójność grafu (mierzoną jako druga co do wielkości wartość własna macierzy sąsiedztwa) prawie tak dobrze jak potęgowanie grafu, zwiększając stopień grafu nie kwadratowo, a jedynie o stałą. 

Przedstawiono również kilka zastosowań tej konstrukcji, m.in. algorytm alternatywny do wyniku O. Reingolda, który pozwala deterministycznie badać osiągalność w grafach nieskierowanych w logarytmicznej pamięci.

Treewidth is an important measure of the “complexity” of a graph, and as part of the Graph Minors project, Robertson and Seymour characterized unavoidable subgraphs of graphs with large treewidth. Here we are interested in unavoidable induced subgraphs instead. In this context, Lozin and Razgon characterized all finite families F of graphs such that F-free graphs have bounded treewidth. I will talk about related result, characterizing which graphs H have the property that excluding H as well as four families of large treewidth (a complete graph, a complete bipartite graph, all subdivisions of a wall, and their line graphs) as induced subgraphs leads to a class of bounded treewidth.

Joint work with Tara Abrishami, Bogdan Alecu, Maria Chudnovsky, and Sepehr Hajebi

We study decision problems for sequences which obey a second-order holonomic recurrence of the form

$f (n + 2) = P (n)f (n + 1) + Q(n)f (n)$

with rational polynomial coefficients, where P is non-constant, Q is non-zero, and the degree of Q is smaller than or equal to that of P . We show that existence of infinitely many zeroes is decidable. We give partial algorithms for deciding the existence of a zero, positivity of all sequence terms, and positivity of all but finitely many sequence terms. If Q does not have a positive integer zero then our algorithms halt on almost all initial values (f (1), f (2)) for the recurrence. We identify a class of recurrences for which our algorithms halt for all initial values. We further identify a class of recurrences for which our algorithms can be extended to total ones.
 

A Majority k-coloring of a directed graph is an assignment of k colors to its vertices in such a way that every vertex has the same color as at most half of its out-neighbors. It is known that every digraph is majority 4-colorable, but it remains an open question whether every digraph is majority 3-colorable. The authors of the paper validate this conjecture for digraphs with a chromatic number at most 6 and digraphs with a dichromatic number at most 3. They also prove analogous theorems for list coloring: digraphs with a list chromatic number at most 6 or list dichromatic number at most 3 are majority 3-choosable. The paper also investigates which digraphs are majority 2-colorable: the authors show that digraphs without directed odd cycles are majority 2-colorable, but in general deciding whether a given digraph is majority 2-colorable is NP-complete. The last result proposed in this paper is proof that every digraph has a fractional majority of 3.9602-coloring.

1. Michael Anastos, Ander Lamaison, Raphael Steiner, Tibor Szabó. Majority Colorings of Sparse Digraphs. Electronic Journal of Combinatorics. 28(2), P2.31. (2021).
2. Kamil Galewski. Majority colorings of sparse digraphs. slides. (2022).

In the basic Lights Out problem, we are given the undirected graph of turned-off lights, and our goal is to turn on all the lights. In the generalized version of this problem, our mission is to assign every vertex a value from 0 to p, such that for every vertex, the sum of values in its neighbors is equal to 0 mod p. The authors not only prove that a generalized version of this problem isn't always solvable but also they show conditions, under which the problem has a solution.

1. János Nagy, Péter Pál Pach. A counterexample to the lights out problem. Journal Graph Theory. 101, 265-273. (2022).
2. Piotr Kaliciak. A counterexample to the lights out problem. slides. (2022).

Mając do dyspozycji zbiór n typów monet o wartościach całkowitych oraz wartość docelową t, w problemie wydawania reszty (change-making) szukamy minimalnej liczby monet, które sumują się do t, zakładając możliwość wykorzystania dowolnej liczby monet każdego typu.

W bardziej ogólnej wersji tego problemu (w wersji all-targets), chcemy obliczyć wyniki dla wszystkich wartości docelowych 0, 1, ..., t. Klasyczny algorytm dynamiczny rozwiązuje ten problem w czasie O(nt).

W publikacji autorzy przedstawiają szereg nowych wyników dotyczących problemu wydawania reszty i innych pokrewnych problemów. Dla u – wartości największej z monet (wagi najcięższego przedmiotu w przypadku problemu plecakowego) pokażemy algorytmy o złożoności:
- Õ(u²+t) dla all-target change-making w oparciu o twierdzenie Erdősa i Grahama dotyczące problemu Frobeniusa;
- Õ(u²+t) dla all-capacities knapsack w oparciu o wcześniejszy algorytm dla change-making;
- Õ(u) dla single-target change-making w oparciu o FFT;
- Õ(nu) dla single-capacity unbounded knapsack, przez wprowadzenie zachłanności umożliwiającej pominięcie obliczania niektórych podproblemów.

Knauer and Valicov showed that multiples conjectures from seemingly different problems all fit into the same framework of cuts in matchings of 3-connected cubic graphs. They unite Tait's, Barnette's, and Tutte's conjectures on Hamiltonicity in cubic graphs, Neumann-Lara's on the dichromatic number of planar graphs, and Hochstättler's on contraction of even digraphs. More precisely, these are all equivalent to conjectures of the form ''Every 3-connected, cubic, bipartite/planar/directed graph contains a perfect matching without (directed) cut''. If you drop two of these restrictions (bipartite, planar, directed), then the conjecture is false. If you drop one or none, then the conjecture remains open. We are investigating the dual version of the conjecture with all three restrictions, namely ''Every directed planar Eulerian triangulation can be vertex-partitioned into two acyclic sets''. This new framework can be useful as a planar Eulerian triangulation has an unique partition into three independent sets.

It was proven by Gallai, that every undirected graph on n vertices contains an induced subgraph on at least n/2 vertices with all degrees even. It is natural to ask a similar question for odd degrees. It was conjectured, that in every graph on n vertices, without isolated vertices, we can find an induced subgraph on at least cn vertices with all degrees odd for some constant c>0. We will prove this conjecture for c=1/10000.

1. Asaf Ferber, Michael Krivelevich. Every graph contains a linearly sized induced subgraph with all degrees odd. Advances in Mathematics. 406, 108534, (2022).
2. Rafał Pyzik. Every graph contains a linearly sized induced subgraph with all degrees odd. slides. (2022).

Since the original statement of Lovas Local Lemma in 1973, multiple variants of the lemma with different levels of complexity have been formulated. We will present a general theorem from which most known variants of LLL follow. Additionally, the results will be generalized to supermodular functions rather than probability measures, allowing a wider range of applications.

1. Dimitris Achlioptas, Kostas Zampetakis. The Lovász Local Lemma is Not About Probability. arXiv:2111.08837. (2021).
2. Justyna Jaworska. The Lovász Local Lemma is Not About Probability. slides. (2022).

Recently we managed with co-authors to settle the complexity of the reachability problem for Vector Addition Systems (VASes) to be Ackermann-complete. Despite of that the combinatorics of VASes still remains mysterious and there is a bunch of very natural problems about which we know shockingly little. The focus of my talk will be on tools. I will present techniques, which led to the proof of Ackermann-hardness for the reachability problem and which hopefully may help in solving the remaining challenges.

We call graph H a minor of graph G, if there exists such a sequence of deletions of vertices, deletions of edges, or contradictions of edges, which transforms G into H. The authors of the paper created a short proof of the result of Kostochka and of Thomasen. The proven theorem states that for every graph whose vertices have the average degree d the graph itself also contains a complete minor of order Ω(d/sqrt(log(d))).

1. Noga Alon, Michael Krivelevich, Benny Sudakov. Complete minors and average degree – a short proof. arXiv:2202.08530. (2022).
2. Jędrzej Kula. Complete minors and average degree – a short proof. slides. (2022).

In the most basic version of the Lamp Lighting Problem, we are given an undirected graph G. We can toggle light in a chosen vertex and all of its neighbors. Our goal is to decide if it is possible to turn on the light in all vertices by performing only moves as described. Authors prove that it is always possible and explore other variants of the problem such as the directed case or the problem of checking if all lighting configurations are possible to achieve.

1. Henrik Eriksson, Kimmo Eriksson, Jonas Sjostrand. Note on the lamp lighting problem. arXiv:math/0411201. (2004).
2. Krzysztof Ziobro. Note on the Lamp Lighting Problem. slides. (2022).

It is a longstanding open problem if posets with a planar cover graph are dim-bounded (meaning that large dimension yields a large standard example as a subposet). This notion is the posets' counterpart of the well-studied χ-boundedness in the graph theory. In my talk, I will focus on summarizing the new progress in this area. The dim-boundedness was recently proved for posests with planar diagram and for posets with planar cover graph and a zero. I will try to sketch some ideas standing behind these results. The other interesting related question in the area is the following. Suppose that a planar poset has a large standard example as a subposet, then, how does this standard example look like? There are two canonical constructions of planar posets with large standard example contained, namely, Kelly's example and Trotter's wheel. We believe that these are (in a structural sense) the only ways to draw a standard example on the plane. For example, we proved that a poset with a planar cover graph, a zero, and large dimension contains a large Trotter's wheel.

The list of coauthors of substantial results that are going to be discussed in my talk: P.Micek, M.Seweryn, H.S.Blake, W.T.Trotter

Let N be a positive integer. A sequence X=(x₁,x₂,…,x_N) of points in the unit interval [0,1) is piercing if {x₁,x₂,…,x_n}∩[i/n,(i+1)/n)≠∅ holds for every n=1,2,…,N and every i=0,1,…,n−1. In 1958 Steinhaus asked whether piercing sequences can be arbitrarily long. A negative answer was provided by Schinzel, who proved that any such sequence may have at most 74 elements. This was later improved to the best possible value of 17 by Warmus, and independently by Berlekamp and Graham. We study a more general variant of piercing sequences. Let f(n)≥n be an infinite nondecreasing sequence of positive integers. A sequence X=(x₁,x₂,…,x_f(N)) is f-piercing if {x₁,x₂,…,x_f(n)}∩[i/n,(i+1)/n)≠∅ holds for every n=1,2,…,N and every i=0,1,…,n−1. A special case of f(n)=n+d, with d a fixed nonnegative integer, was studied by Berlekamp and Graham. They noticed that for each d≥0, the maximum length of any (n+d)-piercing sequence is finite. Expressing this maximum length as s(d)+d, they obtained an exponential upper bound on the function s(d), which was later improved to s(d)=O(d³) by Graham and Levy. Recently, Konyagin proved that 2d⩽s(d)<200d holds for all sufficiently big d. Using a different technique based on the Farey fractions and stick-breaking games, we prove here that the function s(d) satisfies ⌊c₁d⌋⩽s(d)⩽c₂d+o(d), where c₁=ln2/(1−ln2)≈2.25 and c₂=(1+ln2)/(1−ln2)≈5.52. We also prove that there exists an infinite f-piercing sequence with f(n)=γn+o(n) if and only if γ≥1/ln2≈1.44. This is joint work with Marcin Anholcer, Jarosław Grytczuk, Grzegorz Gutowski, Jakub Przybyło, Rafał Pyzik, and Mariusz Zając.

1. Marcin Anholcer, Bartłomiej Bosek, Jarosław Grytczuk, Grzegorz Gutowski, Jakub Przybyło, Rafał Pyzik, Mariusz Zając. On a Problem of Steinhaus. arXiv:2111.01887. (2021).

Intuicyjnie problem Token Sliding możemy rozumieć jako grę, w której otrzymujemy graf oraz żetony ustawione na jego wierzchołkach. Pytamy, czy da się uzyskać zadany stan końcowy poprzez przesuwanie żetonów wzdłuż krawędzi grafu tak, że w żadnym momencie dwa żetony nie łączyła wspólna krawędź.

Formalnie mamy na wejściu graf G oraz zbiory niezależne wierzchołków I_s, I_t i chcemy stwierdzić czy istnieje sekwencja I₁, …, I_s zbiorów niezależnych w G taka, że I₁ = I_s, I_l = I_t oraz I_i ∆ I_i+1 = {u, v} \in E(G).

Wykazano wcześniej, że dla grafów o talii (ang. girth) 4 lub mniejszej problem Token Sliding jest W[1]-trudny. 

Prezentujemy dowód z pracy „Token sliding on graphs of girth five”, że dla grafów o talii 5 lub większej problem Token Sliding  jest fixed-parameter tractable (FPT).

We improve the upper bound for the maximum possible number of stable matchings among n jobs and n applicants (formerly known as n men and n women) from 131072ⁿ to 3.55ⁿ. To establish this bound, we state a novel formulation of a certain entropy bound that is easy to apply and may be of independent interest in counting other combinatorial objects.

Joint work with Cory Palmer

We consider a proof (more accurately, refutation) system based on the Sherali–Adams (SA) operator associated with integer linear programming. If F is a CNF contradiction that admits a Resolution refutation of width k and size s, then we prove that the SA rank of F is ≤ k and the SA size of F is  \leq (k + 1)s + 1. We establish that the SA rank of both the Pigeonhole Principle PHP_n^{n-1} and the Least Number Principle LNP_n  is n − 2. Since the SA refutation system rank simulates the refutation system of Lovász–Schrijver without semidefinite cuts (LS), we obtain as a corollary linear rank lower bounds for both of these principles in LS.

A majority coloring of a directed graph is a vertex coloring in which each vertex has the same color as at most half of its out-neighbors. In this note we simplify some proof techniques and generalize previously known results on various generalizations of majority coloring. In particular, our unified and simplified approach works for paintability - an online analog of list coloring. This is joint work with Marcin Anholcer, Jarosław Grytczuk, Grzegorz Gutowski, Jakub Przybyło, Mariusz Zając.

1. Marcin Anholcer, Bartłomiej Bosek, Jarosław Grytczuk, Grzegorz Gutowski, Jakub Przybyło, Mariusz Zając. Mrs. Correct and Majority Colorings. arXiv:2207.09739. (2022).

Roma to łamigłówka rozgrywana na składającej się z kwadratowych pól planszy rozmiaru n x n. Pola pogrupowane są w obszary składające się z co najwyżej 4 sąsiadujących ze sobą komórek, z których każda albo jest wypełniona, albo ma zostać wypełniona strzałką w jednym z 4 kierunków. Celem gry jest wypełnienie wszystkich komórek strzałkami tak by w każdym obszarze była co najwyżej jedna strzałka w danym kierunku i by podążając zgodnie ze strzałkami można było dojść do wyróżnionego pola Roma z każdego pola na planszy.

Autorzy pracy rozważają złożoność obliczeniową gry i pokazują, że uzupełnienie planszy zgodnie z zasadami jest problemem NP-zupełnym, zliczenie możliwych rozwiązań jest #P zupełne oraz wyznaczenie liczby zadanych z góry strzałek koniecznych by gra miała tylko jedno rozwiązanie jest Σ^P2 - zupełne.

Praca dowodzi też, że zakładając prawdziwość ETH problem uzupełnienia planszy dla danej instancji gry nie może być rozwiązany w czasie O(2^o(n)). Omawia także algorytm programowania dynamicznego rozwiązujący planszę gry, opierający się na strukturach Catalana.

This talk will focus on two graph parameters: stack layouts (aka. book embeddings) and queue layouts of graphs. I will talk about the history of these two graph parameters, their  still not fully understood relationship and some recent breakthroughs.

We study the problem of coloring a unit interval graph that changes dynamically. In our model the unit intervals are added or removed one at a time and have to be colored immediately so that no two overlapping intervals share the same color. After each update, only a limited number of intervals are allowed to be recolored. The limit on the number of recolorings per update is called the recourse budget. In this paper, we show, that if the graph remains k-colorable at all times, and the updates consist of insertions only, then we can achieve the amortized recourse budget of O(k⁷logn) while maintaining a proper coloring with k colors. This is an exponential improvement over the result in [Bosek et al., Recoloring Interval Graphs with Limited Recourse Budget. SWAT 2020] in terms of both k and n. We complement this result by showing the lower bound of Ω(n) on the amortized recourse budget in the fully dynamic setting. Our incremental algorithm can be efficiently implemented. As a byproduct of independent interest, we include a new result on coloring proper circular-arc graphs. Let L be the maximum number of arcs intersecting in one point for some set of unit circular arcs A. We show that if there is a set A′ of non-intersecting unit arcs of size L²−1 such that A∪A′ does not contain L+1 arcs intersecting in one point, then it is possible to color A with L colors. This complements the work on unit circular arc coloring, which specifies sufficient conditions needed to color A with L+1 colors or more. This is joint work with Anna Zych-Pawlewicz.

1. Bartłomiej Bosek, Anna Zych-Pawlewicz. Recoloring Unit Interval Graphs with Logarithmic Recourse Budget. arXiv:2202.08006. (2022).

The talk features a survey as well as recent new results on two independent approaches to derive efficient algorithms for integer programming, namely algorithms based on the geometry of numbers and dynamic programming techniques, with an extra spotlight on the case in which the number of constraints (apart from bounds on the variables) is small. We will highlight open problems and possible future directions.

The presented  results of the speaker have been jointly achieved with Daniel Dadush, Thomas Rithvoss and Robert Weismantel.

It is a long-standing open problem if planar posets are dim-bounded (an analog of chi-bounded in the graph theory). I summarize recent progress on this problem. We explore different notions of what does it mean for posets to be planar. Finally, I will sketch the proof of dim-boundedness in the case of posets with planar cover graphs and a zero.

1. Piotr Micek, Heather C. Smith Blake, William T. Trotter. Boolean dimension and dim-boundedness: Planar cover graph with a zero. arXiv:2206.06942. (2022).
2. Jędrzej Hodor. Dimension of planar posets. slides. (2022).

Erdős and Hajnal proved a general bound: for every H there exists c>0 such that every H-free graph has a clique or stable set of size at least exp(c (log|G|)^1/2). This is still the record for most graphs H, but in some instances one can do better. Let us say H is "friendly" if there exists c>0 such that every H-free graph G has a clique or stable set of size at least exp(c (log|G| loglog|G|)^1/2). We prove that many graphs are friendly – for instance, all split graphs are friendly, and so is the eight-vertex path, and all graphs obtained from a cograph by adding a new vertex joined arbitrarily.

Joint work with Matija Bucić, Tung Nguyen and Alex Scott

In 1989, Nešetřil and Pudlák posed the following challenging question: Do planar posets have bounded Boolean dimension?  We show that every poset with a planar cover graph and a unique minimal element has Boolean dimension at most 13. As a consequence, we are able to show that there is a reachability labeling scheme with labels consisting of O(log n) bits for planar digraphs with a single source. The best known scheme for general planar digraphs uses labels with O(log²n) bits [Thorup, JACM 2004], and it remains open to determine whether a scheme using labels with O(log n) bits exists. The Boolean dimension result is proved in tandem with a second result showing that the dimension of a poset with a planar cover graph and a unique minimal element is bounded by a linear function of its standard example number. However, one of the major challenges in dimension theory is to determine whether dimension is bounded in terms of standard example number for all posets with planar cover graphs.

Within this talk after a quick introduction, I aim to lay out all the ideas behind the proof bounding Boolean dimension.

This is a joint work with Heather Smith Blake and Tom Trotter.

A given pair of two incomparable elements x, y in poset P is called balanced if, of all line extensions P, the element x lies above y by at most 2/3 and on at least 1/3 of all extensions of the poset P. The 1/3 - 2/3 conjecture says that any poset that is not linear has a balanced pair. The talk presents basic definitions and an overview of the most important results in this field.

Feder-Vardi conjecture has been settled independently by Dmitriy Zhuk and Andrei Bulatov. What is perhaps even more interesting, though, is that they not only confirmed the complexity (Feder-Vardi) conjecture, i.e., CSP(B) for a finite structure B is either in P or it is NP-complete, but they also confirmed the algebraic dichotomy conjecture describing  tractable B in terms of operations preserving B.

A similar algebraic dichotomy conjecture called an infinite algebraic dichotomy conjecture has been established for CSP(B) over first-order reducts B of finitely bounded homogeneous structures, all of which are in particular omega-categorical. Despite recent advances towards solving this dichotomy, it still seems to be wide open. One of the reasons is probably that local consistency  and similar algorithmic techniques are in this context not yet fully understood. This step seems to be crucial since the characterization of finite-domain CSP solvable by local consistency  is considered as a major step towards the resolution of the dichotomy.

In this talk, I will survey the results on the local consistency methods in solving CSP and CSP-like problems over omega-categorical structures.

The Tu–Deng Conjecture is concerned with the sum of digits w(n) of n in base 2 (the Hamming weight of the binary expansion of n) and states the following: assume that k is a positive integer and t \in  {1, . . . , 2^k − 2}. Then

#\{ (a, b) ∈ {0, . . . , 2k − 2}^2 : a + b ≡ t mod 2^k − 1, w(a) + w(b) < k \}  \leq ≤ 2^{k-1}

We prove that the Tu–Deng Conjecture holds almost surely in the following sense: the proportion of t \in {1, . . . , 2^k − 2} such that the above inequality holds approaches 1 as k tends to infinity. Moreover, we prove that the Tu–Deng Conjecture implies a conjecture due to T. W. Cusick concerning the sum of digits of n and n + t.

Brooks' Theorem states that every connected graph G with maximum degree d is d-colorable unless G is an odd cycle or a complete graph. It is one of the most famous theorem on graph colorings. In the paper, the author presents yet another proof of this theorem. This proof is based on Alon-Tarsi Theorem and it remains valid in a more general choosability version of Brooks' theorem.

1. Jan Hladký, Daniel Král’, Uwe Schauz. Brooks’ Theorem via the Alon–Tarsi Theorem. Discrete Mathematics. 310 (23), 3426-3428. (2010).
2. Krzysztof Pióro. Brooks’ Theorem via the Alon-Tarsi Theorem. slides. (2022).

A class of graphs is hereditary χ-bounded if it is closed under taking induced subgraphs and every graph’s chromatic number is bounded by some function of its clique number. A well-known recently stated open question has been whether for every hereditary χ-bounded class that function can be chosen to be a polynomial. We provide a counterexample for it; namely, for any function f, we construct a hereditary χ-bounded class containing graphs of large chromatic number. In particular, for any polynomial f, such a class exists, which answers the aforementioned question negatively.

1. Marcin Briański, James Davies, Bartosz Walczak. Separating polynomial χ-boundedness from χ-boundedness. arXiv:2201.08814. (2022).
2. Demian Banakh. Separating polynomial χ-boundedness from χ-boundedness. slides. (2022).

Grafy przecięć geometrycznych to grafy, gdzie wierzchołki odpowiadają figurom geometrycznym w d-wymiarowej przestrzeni euklidesowej. Mogą do to być na przykład kule, kwadraty, hiperkostki. Krawędź między dwoma wierzchołkami istnieje, jeśli dwie figury przecinają się. Jest to typowy sposób modelowania na przykład komunikacji bezprzewodowej.

W pracy autorzy zajmują się obliczaniem średnicy tego typu grafów. Dokładniej rozważają to, czy da się ten problem rozwiązać w czasie poniżej kwadratowym względem liczby wierzchołków. Na referacie zostanie pokazany dowód algorytmu o czasie działania O(n logn) dla sprawdzania, czy średnica jest mniejsza bądź równa 2 dla grafów przecięć kwadratów jednostkowych równoległych do osi. Następnie zostanie pokazane dolne ograniczenie szukania średnicy dla kul jednostkowych na bazie Orthogonal Vectors Hypothesis. Ograniczenie to pokazuje, że nie ma algorytmów pod kwadratowych przy założeniu Orthogonal Vectors Hypothesis.

Consider following problem. We are given a digraph. For every edge, there are 2 options to choose a weight for this edge. We want to pick the weights of edges in a specific way. After picking weights we color vertices. The color of the vertex will be the sum of incoming edges minus the sum of outgoing edges from that vertex. We show that it is always possible to choose weights of edges such that the resulting coloring will be proper. This property is called 2-weight-choosability.

1. Mahdad Khatirinejad, Reza Naserasr, Mike Newman, Ben Seamone, Brett Stevens. Digraphs are 2-Weight Choosable. Electronic Journal of Combinatorics. 18(1), P21. (2011).
2. Bartosz Podkanowicz. Digraphs are 2-weight choosable. slides. (2022).

A graph G is k-list-critical if it is not (k-1)-choosable, but every proper subgraph of G is (k-1)-choosable. We give a new lower bound for the average degree of incomplete k-list-critical graphs and online k-list-critical graphs. The presented bound improves the earlier known lower bounds for k = 4,5,6.

1. Hal Kierstead, Landon Rabern. Extracting list colorings from large independent sets. arXiv:1512.08130. (2015).
2. Landon Rabern. A better lower bound on average degree of 4-list-critical graphs. arXiv:1602.08532. (2016).
3. Łukasz Selwa. A better lower bound on average degree of 4-list-critical graphs. slides. (2022).

We establish a list of characterizations of bounded twin-width for hereditary classes of totally ordered graphs: as classes of at most exponential growth studied in enumerative combinatorics, as monadically NIP classes studied in model theory, as classes that do not transduce the class of all graphs studied in finite model theory, and as classes for which model checking first-order logic is fixed-parameter tractable studied in algorithmic graph theory.

This has several consequences.First, it allows us to show that every hereditary class of ordered graphs either has at most exponential growth, or has at least factorial growth. This settles a question first asked by Balogh, Bollobás, and Morris [Eur. J. Comb. '06] on the growth of hereditary classes of ordered graphs, generalizing the Stanley-Wilf conjecture/Marcus-Tardos theorem. Second, it gives a fixed-parameter approximation algorithm for twin-width on ordered graphs. Third, it yields a full classification of fixed-parameter tractable first-order model checking on hereditary classes of ordered binary structures. Fourth, it provides a model-theoretic characterization of classes with bounded twin-width.

Those results are joint work with Bonnet, Giocanti, Ossona de Mendez, Simon, Thomasse, accepted to STOC'22.

Time permitting, I will also discuss the more general landscape of monadically NIP graph classes, generalizing both nowhere dense classes and classes of bounded twin-width.

An online chain partitioning algorithm is presented with one element of a poset at a time and has to assign it to a chain, partitioning the poset. We consider posets with elements represented by unit length intervals. The paper slightly improves the lower bound for the minimum number of chains needed by an online algorithm to partition these posets from ⌊3/2 w⌋ to ⌈3/2 w⌉.

1. Csaba Biró, Israel R. Curbelo. Improved lower bound on the on-line chain partitioning of semi-orders with representation. arXiv:2111.04790. (2021).
2. Rafał Kilar. Lower Bounds on the On-line Chain Partitioning of Semi-orders with Representation. slides. (2022).

Simple unit-Monge matrices form a special subclass of square matrices, which can be represented implicitly in linear space by permutations. Somewhat surprisingly, the subclass is closed under distance multiplication. We will show connection between simple unit-Monge matrices and seaweed braids: braids in which each pair of strings crosses at most once. In particular, distance multiplication is equivalent to a "combing procedure", where double-crossings in braid are removed. We will discuss applications of these methods to a few subsequence problems. In particular, the combing procedure can be exploited to obtain an elegant algorithm for all-substring LCS problem.

1. Alexander Tiskin. Semi-local string comparison: algorithmic techniques and applications. arXiv:0707.3619. (2007).
2. Krzysztof Potępa. Unit-Monge matrices and seaweed braids. slides. (2022).

Strukturalne parametry grafów, takie jak treewidth, pathwidth i clique-width, są głównym tematem badań sparametryzowanej złożoności. Głównym celem badań w tej dziedzinie jest zrozumienie „ceny ogólności” tych szerokości: kiedy przechodzimy od pojęć bardziej restrykcyjnych do bardziej ogólnych, jakie są problemy, w których złożoność pogarsza się z fixed-parameter tractable do intractable? Ten rodzaj pytania jest już bardzo dobrze zbadany, ale, co jest dość uderzające, algorytmiczna granica między dwoma (prawdopodobnie) najbardziej centralnymi pojęciami szerokości (notacjami), treewidth i pathwidth, jest nadal niezrozumiała: obecnie nie jest znany żaden naturalny problem na grafie, który byłby W-trudny dla jednego, ale FPT dla drugiego. Rzeczywiście, zaskakującym rozwojem ostatnich kilku lat była obserwacja, że: w przypadku wielu najbardziej paradygmatycznych problemów ich złożoność dla tych dwóch parametrów w rzeczywistości dokładnie się pokrywają, pomimo faktu, że szerokość drzewa jest parametrem o wiele bardziej ogólnym. W ten sposób wydaje się, że dodatkowa ogólność szerokości drzewa nad szerokością ścieżki często przychodzi „za darmo”.

Naszym głównym wkładem w ten artykuł jest odkrycie pierwszego naturalnego przykładu, w którym ta ogólność ma wysoką cenę. Rozważamy Grundy Coloring, wariację kolorowania, w której próbujemy obliczyć najgorsze możliwe kolorowanie, które można przypisać do grafu przez zachłanny algorytm First-Fit . Pokazujemy, że ten dobrze zbadany problem jest parametryzowany (FPT) przez pathwidth; jednakże to staje się znacznie trudniejsze (W[1]-hard), gdy jest sparametryzowany przez treewidth. Ponadto pokazujemy, że Grundy Coloring sprawia, że jest drugi skok złożoności dla bardziej ogólnych szerokości, gdy staje się para-NP-hard dla clique-width. Dlatego Grundy Coloring ładnie oddaje złożoność kompromisów między trzema najlepiej zbadanymi parametrami. Uzupełniając obraz, pokazujemy, że Grundy Coloring jest parametryzowane przez FPT według modular-width.

This talk focuses on Eulerian graphs whose arcs are directed and labelled in a group. Each circuit yields a word over the group, and a circuit is non-zero if this word does not evaluate to 0. We give a precise min-max theorem for the following problem. Given a vertex v, what is the maximum number of non-zero circuits in a circuit-decomposition where each circuit begins and ends at v?

This is joint work with Jim Geelen and Paul Wollan. Our main motivation is a surprising connection with vertex-minors which is due to Bouchet and Kotzig.

Nash-Williams theorem (tree-packing theorem) is a classical result due to Nash-Williams (1961) that characterizes graphs with k edge-disjoint spanning trees. In the seminar, I will present a short and elegant proof of the theorem.

1. Boliong Chen, Makoto Matsumoto, Jianfang Wang, Zhongfu Zhang, Jianxun Zhang. A short proof of Nash-Williams' theorem for the arboricity of a graph. Graphs and Combinatorics, 10 (1). 27-28. (1994).
2. Jacek Salata. A Short Proof of Nash-Williams' Theorem for the Arboricity of a Graph. slides. (2022).

The hat guessing game is a game in which bears sit in the vertices of an undirected graph. A demon puts hats on the bears' heads. Each hat has one of the h available colors. Each bear sees only the hat colors of his neighbors. The goal of the bears is to guess the color of their hats - each bear has g tries to guess his hat color. The bears win if at least one of them has guessed the color of his hat correctly. This paper describes the relationship between the hat guessing game and the independence polynomial of graphs.

1. Václav Blažej, Pavel Dvořák, Michal Opler. Bears with Hats and Independence Polynomials. arXiv:2103.07401. (2021).
2. Kamil Galewski. Bears with Hats and Independence Polynomials. slides. (2022).

An n-wall is a graph obtained from the square grid with n rows and 2n columns by deleting every odd vertical edge in every odd row and even vertical edge in every even row, then deleting the two resulting vertices of degree 1, and finally subdividing each edge any number of times. Thomassen showed that there exists a function f(n,m) such that every f(n,m)-wall contains an n-wall such that every path between two branch vertices has length divisible by m. We study the asymptotics of the optimal such function f(n,m). For odd m we show that f(n,m) = O(n·poly(m)). In the case m=2, we obtain a bound f(n, 2) = O(n·log n).

This is joint work with Piotr Micek, Raphael Steiner and Sebastian Wiederrecht.

Consider G(n, p) to be the probability space of random graphs on n vertices with edge probability p. We will be considering subsets of this space defined by monotone graph properties. A monotone graph property P is a property of graphs such that

a) P is invariant under graph automorphisims.
b) If graph H has property P , then so does any graph G having H as a subgraph.
 

A monotone symmetric family of graphs is a family defined by such a property. One of the first observations made about random graphs by Erdos and Renyi in their seminal work on random graph theory [12] was the existence of threshold phenomena, the fact that for many interesting properties P , the probability of P appearing in G(n, p) exhibits a sharp increase at a certain critical value of the parameter p. Bollob ́as and Thomason proved the existence of threshold functions for all monotone set properties ([6]), and in [14] it is shown that this behavior is quite general, and that all monotone graph properties exhibit threshold behavior, i.e. the probability of their appearance increases from values very close to 0 to values close to 1 in a very small interval. More precise analysis of the size of the threshold interval is done in [7]. This threshold behavior which occurs in various settings which arise in combinatorics and computer science is an instance of the phenomenon of phase transitions which is the subject of much interest in statistical physics. One of the main questions that arises in studying phase transitions is: how “sharp” is the transition? For example, one of the motivations for this paper arose from the question of the sharpness of the phase transition for the property of satisfiability of a random k-CNF Boolean formula. Nati Linial, who introduced me to this problem, suggested that although much concrete analysis was being performed on this problem the best approach would be to find general conditions for sharpness of the phase transition, answering the question posed in [14] as to the relation between the length of the threshold interval and the value of the critical probability.

A contact graph of a ball packing is a graph with non-intersecting balls as vertices and edges between pairs of tangent balls. In the seminar, we will focus on the upper bounds for the average degree of such graphs in any number of dimensions.

1. Alexey Glazyrin. Contact graphs of ball packings. arXiv:1707.02526. (2017).
2. Szymon Salabura. Contact Graphs on Ball Packings. slides. (2022).

It has been conjectured that every planar triangle-free graph G has exponentially many proper vertex-3-colorings. In this paper, the conjecture is disproved. It is also shown that the conjecture holds if we add an assumption about the non-existence of separating cycles of lengths 4 and 5. Specifically, it is proved that the number of proper vertex-3-colorings of every triangle-free planar graph with n vertices and with no separating cycle of length 4 or 5 is at least 2^n/17700000.

1. Carsten Thomassen. Exponentially many 3-colorings of planar triangle-free graphs with no short separating cycles. Journal of Combinatorial Theory, Series B. (2021).
2. Mateusz Pach. Exponentially many 3-colorings of planar graphs with no short separating cycles. slides. (2022).

W problemie odległości między drzewami dane są dwa ukorzenione drzewa z etykietami na krawędziach. Dodatkowo dla każdego wierzchołka jego dzieci mają ustalony porządek. Naszym celem jest znalezienie minimalnej liczby operacji kontrakcji krawędzi i zmiany etykiety krawędzi, tak aby doprowadzić oba drzewa do takiego samego drzewa.

Autor pracy pokazuje algorytm o złożoności O(n^2.9546) dla wariantu tego problemu, w którym operacje mają koszty jednostkowe. Jest to pierwszy podsześcienny algorytm dla problemu odległości edycyjnej między drzewami. Warto tutaj zwrócić uwagę, że dla wariantu o dowolnych kosztach operacji istnieje warunkowe ograniczenie dolne, które mówi, że nie istnieje dla tego problemu algorytm podsześcienny. Zatem autor pokazuje, że wariant z jednostkowymi kosztami najprawdopodobniej jest istotnie prostszy od wariantu ogólnego.

Aby złamać granicę O(n³) autor redukuje problem do mnożenia macierzy max-plus, w których sąsiednie elementy różnią się co najwyżej o stałą. O takim problemie zostało już udowodnione wcześniej, że może zostać rozwiązany w czasie podsześciennym. 

How many edges do we need to build a k-uniform hypergraph that cannot be properly two colored? Using the probabilistic argument Erdös proved in 1964, that there exist such hypergraphs with roughly k²·2^k edges. However, without a random source at hand, the sizes that we can achieve by efficient procedures are much larger. The first and only known explicit construction with 2^k+o(k) edges was proposed by Gebauer in 2013. We will discuss how it can be improved first by randomizing and then derandomizing it once more.

The paper considers two classes of planar graphs: maximal planar graphs and maximal bipartite planar graphs. The authors describe how these graphs can be oriented in the way that each vertex has prescribed indegree. Then the relation of such orientations to specific graph decompositions and orderings on the vertex set is provided. Discussed orientations can be used to characterize some of the planar graphs. Described properties have applications e.g. in graph drawing and planar augmentation problems.

1. Hubert de Fraysseix, Patrice Ossona de Mendez. On topological aspects of orientations. Discrete Mathematics. 229(1-3). 57-72. (2001).
2. Karolina Gontarek. On topological aspects of orientations. slides. (2022).

By describing a winning strategy for Mrs. Correct in the coloring game of Mr. Paint and Mrs. Correct author presents a purely combinatorial proof of a strengthening of Alon and Tarsi's Theorem. Strengthening of the theorem also leads to the strengthening of its applications, for example, upper bounds for list chromatic numbers of bipartite graphs, list chromatic indices of complete graphs, and chess tournament time scheduling problem with unreliable participants.

1. Uwe Schauz. Flexible Color Lists in Alon and Tarsi's Theorem, and Time Scheduling with Unreliable Participants. Electronic Journal of Combinatorics. 17. R13. (2010).
2. Ruslan Yevdokymov. Flexible Color Lists in Alon and Tarsi’s Theorem. slides. (2022).

A simple topological graph is a graph drawn in the plane so that its vertices are represented by points, and its edges are represented by non-self-intersecting arcs connecting the corresponding points, with the property that any two edges have at most one point in common. In 2003, Pach-Solymosi-Tóth showed that every n-vertex complete simple topological graph contains a topological subgraph on m = Ω(log n) vertices that is weakly isomorphic to the complete convex geometric graph or to the complete twisted graph on m vertices. Here,  we improve this bound to  (log n)^1/4−o(1). I will also discuss other related problems as well.

This is joint work with Ji Zeng.

Consider G(n, p) to be the probability space of random graphs on n vertices with edge probability p. We will be considering subsets of this space defined by monotone graph properties. A monotone graph property P is a property of graphs such that

a) P is invariant under graph automorphisims.
b) If graph H has property P , then so does any graph G having H as a subgraph.
 

A monotone symmetric family of graphs is a family defined by such a property. One of the first observations made about random graphs by Erdos and Renyi in their seminal work on random graph theory [12] was the existence of threshold phenomena, the fact that for many interesting properties P , the probability of P appearing in G(n, p) exhibits a sharp increase at a certain critical value of the parameter p. Bollob ́as and Thomason proved the existence of threshold functions for all monotone set properties ([6]), and in [14] it is shown that this behavior is quite general, and that all monotone graph properties exhibit threshold behavior, i.e. the probability of their appearance increases from values very close to 0 to values close to 1 in a very small interval. More precise analysis of the size of the threshold interval is done in [7]. This threshold behavior which occurs in various settings which arise in combinatorics and computer science is an instance of the phenomenon of phase transitions which is the subject of much interest in statistical physics. One of the main questions that arises in studying phase transitions is: how “sharp” is the transition? For example, one of the motivations for this paper arose from the question of the sharpness of the phase transition for the property of satisfiability of a random k-CNF Boolean formula. Nati Linial, who introduced me to this problem, suggested that although much concrete analysis was being performed on this problem the best approach would be to find general conditions for sharpness of the phase transition, answering the question posed in [14] as to the relation between the length of the threshold interval and the value of the critical probability.

In this seminar, we will talk about Maryam Mirzakhani, who had an enormous influence on research about Combinatorics. We will study her idea of creating a small (with (only!) 63 vertices), non-4-choosable planar graph, which is also 3-choosable. We will also consider other problems she worked on.

1. William J. Martin. On an early paper of Maryam Mirzakhani. arXiv:1709.07540. (2017).
2. Wojciech Buczek. On an early paper of Maryam Mirzakhani. slides. (2022).

Word creation from lists of size t is a problem where for alphabet Σ each sign of created word is chosen from a list of t different signs from Σ. Word is "square-free" when it does not contain any squares. A square is a word of form ww with w being a nonempty word. The author first shows that there are at least 2.45ⁿ square-free words of length n created from lists of 4. This is an improvement from the previous bound which is 2ⁿ. After that, the main result of the paper is shown which is an existence of at least 1.25ⁿ words of length n from lists of 3.

1. Rosenfeld, Matthieu. Avoiding squares over words with lists of size three amongst four symbols. arXiv:2104.09965. (2021).
2. Maciej Nemś. Avoiding squares over words with lists of size three amongst four symbols. slides. (2022).

A (strict) bramble in a graph G is a collection of subgraphs of G such that the union of any number of them is connected. The order of a bramble is the smallest size of a set of vertices that intersects each of the subgraphs in it. Brambles have long been part of the graph minor theory toolkit, in particular, because a bramble of high order is an obstruction to existence of  a low width tree decomposition.

We will discuss two dimensional analogues of brambles. In particular, we show that a bramble of high order in a two-dimensional simplicial complex X is an obstruction to existence of a low multiplicity continuous map from X  to the plane (and more generally to any two-dimensional collapsible complex). As an application, we show that strong products of three paths have unbounded stack number.

Based primarily on joint work with David Eppstein,  Robert Hickingbotham, Laura Merker, Michał T. Seweryn and David R. Wood. (https://arxiv.org/abs/2202.05327)

We consider some coloring issues related to the famous Erdős Discrepancy Problem. A set of the form A_s,_k={s,2s,…,ks}, with s,k ∈ N, is called a homogeneous arithmetic progression. We prove that for every fixed k there exists a 2-coloring of N such that every set A_s,k is perfectly balanced (the numbers of red and blue elements in the set A_s,k differ by at most one). This prompts reflection on various restricted versions of Erdős' problem, obtained by imposing diverse confinements on parameters s,k. In a slightly different direction, we discuss a majority variant of the problem, in which each set A_s,k should have an excess of elements colored differently than the first element in the set. This problem leads, unexpectedly, to some deep questions concerning completely multiplicative functions with values in {+1,−1}. In particular, whether there is such a function with partial sums bounded from above.

We prove that for every graph F with at least one edge there is a constant c_F and there are graphs H of arbitrarily large chromatic number and the same clique number as F such that every F-free induced subgraph of H has chromatic number at most c_F. (Here a graph is F-free if it does not contain an induced copy of F.) This generalizes theorems of Briański, Davies and Walczak, and of Carbonero, Hompe, Moore and Spirkl.  We further show an analogous statement where clique number is replaced by odd girth.

Joint work with Antonio Girao, Freddie Illingworth, Emil Powierski, Michael Savery, Youri Tamitegama and Jane Tan.

Upon gaining p% votes in an election in a proportional system, a party appoints p% of its proposed candidates to represent the party. The order of candidates to appoint is chosen beforehand. This may create tensions if the party members are not perfectly aligned politically, if some candidates of particular tendency are lower down the list and thus less likely to be appointed. This presentation examines the problem of existence and characterization of the list that would not create such tension and related problems.

1. Amoz Kats. Can a Party Represent Its Constituency?. Public Choice. 44(3), 453-456. (1984).
2. Marcin Serwin. Can a party represent its constituency? slides. (2022).

The author is considering a planar graph, where every vertex has a list of 2 colors, and every coloring of this graph has to choose for every vertex one of these two colors. Unlike the standard colorings, the author doesn't mind having a monochromatic edge, but he tries to avoid having a monochromatic triangle. In this paper, he not only proves, that every planar graph can be colored this way, for every list assignment, but also he proves a stronger result for graphs with some vertices pre-colored.

1. Carsten Thomassen. 2-List-coloring planar graphs without monochromatic triangles. Journal of Combinatorial Theory, Series B. 98(6). 1337-1348. (2008).
2. Piotr Kaliciak. List coloring planar graphs without monochromatic triangles. slides. (2022).

If a graph contains no large complete subgraph but nonetheless has high chromatic number what can we say about the structure of such a graph? This question naturally leads to investigation of χ-bounded classes of graphs — graph classes where a graph needs to contain a large complete subgraph in order to have high chromatic number. This an active subfield of graph theory with many long standing open problems as well as interesting recent developments.

In this talk I will present a construction of a hereditary class of graphs which is χ-bounded but not polynomially χ-bounded. This construction provides a negative answer to a conjecture of Esperet that every χ-bounded hereditary class of graphs is polynomially χ-bounded. The construction is inspired by a recent paper of Carbonero, Hompe, Moore, and Spirkl which provided a counterexample to another conjecture of Esperet.

This is joint work with James Davies and Bartosz Walczak (available at arXiv:2201.08814)

An outerplanar graf is a planar graph whose vertices can all be drawn on the outer face. The author researched the problem of coloring outerplanar graphs from lists. First, it is shown that the outerplanar graph is L-colorable if satisfies |L(v)| ≥ min{deg(v),4} and is bipartite. Then additional assumptions are searched for so that a similar inequality could define L-colorability in general outerplanar graphs. The results given by the author are the best possible for each condition in the bounds and hypotheses.

1. J.P. Hutchinson. On list-coloring outerplanar graphs. J. Graph Theory. 59, 59-74. (2008).
2. Katarzyna Król. On List-Coloring Outerplanar Graphs. slides. (2022).

Since every planar graph G can be colored by 4 colors, there is also an integer m such that G is (4m,m)-choosable. The problem here is that such m is changing with G. The author of this paper proves that there cannot be such a universal m that every planar graph is (4m,m)-choosable. To be precise he shows that for each positive integer m, there is a planar graph G which is not (4m+⌊(2m-1)/9⌋,m)-choosable. Also, he poses some conjectures about planar graphs multiple list coloring.

1. Xuding Zhu. Multiple list colouring of planar graphs. Journal of Combinatorial Theory, Series B. 122, 794-799. (2017).
2. Jędrzej Kula. Multiple list colouring of planar graphs. slides. (2022).

In this talk, I will present some recent results on two variants of Hadwiger's conjecture.

             First, I will discuss the so-called Odd Hadwiger's conjecture, a strengthening of Hadwiger's conjecture proposed by Gerards and Seymour in 1995. First I will show how, using a simple new trick, one can reduce the problem of coloring graphs with no odd K_t-minor to coloring graphs with no K_t-minor up to a constant factor of 2, thereby improving previous upper bounds for this problem. Then, I will outline how a similar idea can be used to significantly improve the known bounds for clustered colorings of odd K_t-minor free graphs, in which we look for (possibly improper) colorings with monochromatic components of small size.

            Second,  I will discuss the so-called List Hadwiger's conjecture,  which states that there exists a constant c such that every graph with no K_t-minor is ct-choosable (i.e.,  list colorable).   I will show a probabilistic construction of a new lower bound  2t-o(t)  for list coloring  K_t-minor free graphs,  this refutes a conjecture by Kawarabayashi and Mohar which stated that one can take c=3/2.  I will then show how some well-chosen modifications to our construction can be used to prove lower bounds also for list coloring  H-minor free graphs where  H  is non-complete. In particular, I will show that   K_s,t-minor free graphs  for large comparable  s  and  t  can have list chromatic number  at least  (1-o(1))(s+t+min(s,t)),  this refutes a 2001 conjecture by Woodall.

Hadwiger conjecture states, that for every K_s+1 minor free graph it can be colored with s colors. For now, it is wide open. There are plenty of well-known results improving the bound on the number of colors. However, there exists another approach to make the problem easier. We can relax the notion of proper coloring. A graph class can be η-clustered colored with n colors if in every graph only n colors are used and the size of each monochromatic component is bounded by η. Liu and Wood proved that for a class of graphs excluding K_s+1 minor can be η(s)-clustered colored with s+2 colors, which is almost optimal (s < s+2). I will describe their approach and prove the result in a simplified case.

1. Chun-Hung Liu, David R. Wood. Clustered Coloring of Graphs Excluding a Subgraph and a Minor. arXiv:1905.09495. 2019.
2. Jędrzej Hodor. Clustered Coloring of Graphs Excluding a Subgraph and a Minor. slides. (2022).

A path of length 2n, that starts in (0,0) and at each step moves from (x,y) to (x+1,y+1) or (x+1,y-1) is a Dyck path, if it ends in (2n,0) and never passes below y=0 line. Such paths are counted by Catalan numbers. In this presentation, we'll show, that the Dyck paths for fixed n form a matroid. We'll show what are bases, independent sets, and other matroid-related terms in this object, explore some properties of this matroid, and see how it generalizes to shifted matroids.

1. Federico Ardila. The Catalan matroid. arXiv:math/0209354. 2002.
2. Grzegorz Gawryał. The Catalan Matroid. slides. (2022).

We study the Maximum Independent Set of Rectangles (MISR) problem, where we are given a set of axis-parallel rectangles in the plane and the goal is to select a subset of non-overlapping rectangles of maximum cardinality. In a recent breakthrough, Mitchell [2021] obtained the first constant-factor approximation algorithm for MISR. His algorithm achieves an approximation ratio of 10 and it is based on a dynamic program that intuitively recursively partitions the input plane into special polygons called corner-clipped rectangles (CCRs), without intersecting certain special horizontal line segments called fences.

In this talk, I will present a (2+ϵ)-approximation algorithm for MISR which is also based on a recursive partitioning scheme. First, we use a partition into a class of axis-parallel polygons with constant complexity each that are more general than CCRs. This allows us to provide an arguably simpler analysis and at the same time already improves the approximation ratio to 4. Then, using a more elaborate charging scheme and a recursive partitioning into general axis-parallel polygons with constant complexity, we improve our approximation ratio to 2+ϵ. In particular, we construct a recursive partitioning based on more general fences which can be sequences of up to O(1/ϵ) line segments each. This partitioning routine and our other new ideas may be useful for future work towards a PTAS for MISR.

The factor of a graph is its spanning subgraph which adheres to given constraints on the degrees. The authors of the article discuss the f-factor, which for every vertex defines a set of possible degrees. The main result shows a new sufficient condition for the existence of an f-factor in a given graph. Authors obtain it by using Combinatorial Nullstellensatz.

1. Hamed Shirazi, Jacques Verstraëte. A Note on Polynomials and f-Factors of Graphs. Electronic journal of combinatorics. 15, N22. 2008.
2. Krzysztof Ziobro. A note on polynomials and f -factors of graphs. slides. (2022).

Problemy sortowania od długiego czasu są obiektem różnego rodzaju badań. Ostatnio dwie gry na telefon w tematyce sortowania zyskały na popularności. W tych grach, gracz ma do dyspozycji urny wypełnione kolorowymi obiektami (w przypadku jednej są to kule, a w przypadku drugiej woda) oraz kilka pustych urn, a jego celem jest posortowanie obiektów zgodnie z kolorami. W jednym ruchu może on przenieść obiekty z jednej urny do drugiej, jeżeli kolor przenoszonych obiektów zgadza się z kolorem najwyższego obiektu docelowej urny lub urna ta jest pusta.

W pracy autorzy badają złożoność obliczeniową tych łamigłówek. Na początku pokazują, że te gry są w równoważne pod kątem rozwiązywalności. Dokładniej mówiąc, rozwiązywalność stanu początkowego gry nie zależy od tego czy obiekty zachowują się jak kule, czy jak woda. Dowodzą również, że dla każdej tak-instancji istnieje rozwiązanie wielomianowego rozmiaru, co pokazuje, że problem rozwiązywalności tych łamigłówek jest w NP. Następnie uzasadniają, że ten problem jest NP-zupełny. Znajdują również wielomianowe algorytmy dla szczególnych przypadków. Na samym końcu zastanawiają się, jak wiele pustych urn jest potrzebnych, aby instancja była rozwiązywalna niezależnie od początkowego rozmieszczenia obiektów. Pokazują nietrywialne ograniczenia (dolne i górne) zależne od ilości początkowo zapełnionych urn i ich pojemności.

We prove that for every t∈ℕ there is a constant γ(t) such that every graph with twin-width at most t and clique number ω has chromatic number bounded by 2^{γ(t) log^{4t+3} ω}. In other words, we prove that graph classes of bounded twin-width are quasi-polynomially χ-bounded. This provides a significant step towards resolving the question of Bonnet et al. [ICALP 2021] about whether they are polynomially χ-bounded.

This is a joint work with Michał Pilipczuk

A k-vertex set S of vertices in a planar graph G is free if, for any k-point set X, there exists a non-crossing straight-line drawing of G with the vertices of S mapped to the points in X.  In this talk we survey the history and applications of free sets and present two recent results [1,2]:

1. Free sets and collinear sets: If G has any drawing in which all vertices of S appear on a line, then S is a free set.

2. Free sets and dual circumference:  If G has bounded degree, then the size of the largest collinear set in G is proportional to the length of the longest cycle in the dual of G.

[1] Vida Dujmović, Fabrizio Frati, Daniel Gonçalves, Pat Morin, and Günter Rote: Every collinear set in a planar graph is free. Discrete & Computational Geometry, 65:999–1027, 2021.

[2] Vida Dujmovic, Pat Morin: Dual Circumference and Collinear Sets. SoCG 2019: 29:1-29:17.

We study the problem of online facility location with delay. In this problem, a sequence of n clients appear in the metric space, and they need to be eventually connected to some open facility. The clients do not have to be connected immediately, but such a choice comes with a penalty: each client incurs a waiting cost (the difference between its arrival and connection time). At any point in time, an algorithm may decide to open a facility and connect any subset of clients to it. This is a well-studied problem both of its own, and within the general class of network design problems with delays.
Our main focus is on a new variant of this problem, where clients may be connected also to an already open facility, but such action incurs an extra cost: an algorithm pays for waiting of the facility (a cost incurred separately for each such "late" connection). This is reminiscent of online matching with delays, where both sides of the connection incur a waiting cost. We call this variant two-sided delay to differentiate it from the previously studied one-sided delay.
We present an O(1)-competitive deterministic algorithm for the two-sided delay variant. On the technical side, we study a greedy strategy, which grows budgets with increasing waiting delays and opens facilities for subsets of clients once sums of these budgets reach certain thresholds. Our technique is a substantial extension of the approach used by Jain, Mahdian and Saberi [STOC 2002] for analyzing the performance of offline algorithms for facility location.
We then show how to transform our O(1)-competitive algorithm for the two-sided delay variant to O(logn/loglogn)-competitive deterministic algorithm for one-sided delays. We note that all previous online algorithms for problems with delays in general metrics have at least logarithmic ratios.

Joint work with Marcin Bienkowski, Martin Böhm and Jan Marcinkowski

We prove that the Alon-Tarsi number of a planar graph is less or equal to 5. Alon Tarsi number is an important parameter for the graph. It is greater than the choice number and paintability for every graph. We show the modification of the standard argument presented by Carsten Thomassen. We construct a special orientation that doesn't have Euler subgraphs and allows us to reason about the Alon-Tarsi number.

1. X. Zhu. The Alon-Tarsi number of planar graphs. arXiv:1711.10817. (2017).
2. Bartosz Podkanowicz. Thomassen orientations. slides. (2022).

The class NC represents the problems that have efficient parallel algorithms. In this work, the authors present two algorithms. The first algorithm proves that the perfect matching problem for bipartite graphs is in quasi-NC². The second algorithm proves that the same problem is in the RNC class and uses only O(log² n) random bits. Note that a complete derandomization would be achieved when the number of random bits comes down to O(log n).

1. S. A. Fenner, R. Gurjar, T. Thierauf. Bipartite Perfect Matching is in quasi-NC. arXiv:1601.06319. (2016).
2. Jędrzej Kula. Bipartite Perfect Matching is in quasi-NC. slides. (2022).

In the game coloring game two players are given graph and a set of k colors. Players take turns, coloring properly an uncolored vertex. The goal of the first player is to complete the coloring of the graph, while the other one tries to prevent him from achieving it. The game chromatic number of a graph is the minimum number of colors needed for the first player to win. In this presentation I will show bounds for the game chromatic number for some classes of graphs.

1. Krzysztof Pióro. Graph Coloring Game. slides. (2022).

We obtain exact expressions counting the satisfiable 2-SAT formulae and describe the structure of associated implication digraphs. Our approach is based on generating function manipulations. To reject the combinatorial specificities of the implication digraphs, we introduce a new kind of generating function, the Implication generating function, inspired by the Graphic generating function used in digraph enumeration. Using the underlying recurrences, we make accurate numerical predictions of the phase transition curve of the 2-SAT problem inside the critical window. We expect these exact formulae to be amenable to rigorous asymptotic analysis using complex analytic tools, leading to a more detailed picture of the 2-SAT phase transition in the future.

In the Hats game, the sages, located at graph vertices, try to guess colors of their own hats, only seeing colors of hats on sages at the adjacent vertices. If using a deterministic strategy, at least one sage can guess the color of his own hat correctly, we say that the sages win. In this presentation, we consider the hat guessing number - the maximum number of possible colors, for which the sages can guarantee the win. We will see examples of graphs contradicting the previously stated hypothesis, that the hat guessing number is smaller than the graph's maximal degree + 1. We also show its independence from the graph's diameter.

1. Aleksei Latyshev, Konstantin Kokhas. The Hats game. On max degree and diameter. arXiv:2108.08065. (2021).
2. Szymon Salabura. The Hats game. On max degree and diameter. slides. (2021).

A constellation is a finite collection of circles in the plane in which no three circles are tangent at the same point. In 1959, Ringel asked how many colours are required to colour the circles of a constellation so that tangent circles receive different colours. We present a solution to Ringel's circle problem by constructing constellations that require arbitrarily many colours.

Joint work with Chaya Keller, Linda Kleist, Shakhar Smorodinsky, and Bartosz Walczak

We prove that there are 3-CNF formulas over n variables that can be refuted in resolution in width w but require resolution proofs of size $n^{\Omega (w)}$. This shows that the simple counting argument that any formula refutable in width w must have a proof in size  $n^{O (w)}$ is essentially tight. Moreover, our lower bound generalizes to polynomial calculus resolution (PCR) and Sherali-Adams, implying that the corresponding size upper bounds in terms of degree and rank are tight as well. The lower bound does not extend all the way to Lasserre, however, since we show that there the formulas we study have proofs of constant rank and size polynomial in both n and w.

Thomassen showed in 1994 that all planar graphs are 5-choosable and Škrekovski showed in 1998 that all K₅-minor-free graphs also are 5-choosable. In this presentation we will take a look at two different proofs of the latter theorem: the Škrekovski's one from the original paper, and the one proposed by Wenjie Hea, Wenjing Miao and Yufa Shenb in 2007.

1. Wenjie He, Wenjing Miao, Yufa Shen. Another proof of the 5-choosability of -minor-free graphs. Discrete Mathematics. 308(17), 4024-4026. (2008).
2. Kacek Salata. Choosability of K5-minor-free graphs. slides. (2022).

The well-known open Hadwiger's conjecture asserts that every simple graph G which is not t-colorable has K_t+1 minor. In this presentation, we will take a look at the proof of a relaxed version of this conjecture (in terms of so-called "defective colorings" - i.e. allowing a "small" number of monochromatic edges), as well as see how it can be useful for solving some other graph problems.

1. Katherine Edwards, Dong Yeap Kang, Jaehoon Kim, Sang-il Oum, Paul Seymour. A relative of Hadwiger's conjecture. arXiv:1407.5236. (2014).
2. Demian Banakh. A Relative of Hadwiger’s Conjecture. slides. (2022).

In this talk, inspired by a "17-points" problem of Steinhaus (Problems 6 and 7 from his book "Sto zadań"), we discuss infinite sequences of real numbers in [0,1).
For a function f:N--->N, we say that a sequence X is f-piercing if for every finite m, the first f(m) elements of X contain at least one element in every interval [i/m,(i+1)/m) for i from 0 up to m-1. There is a nice construction of (m/ln2)-piercing sequence due to de Bruijn and Erdős which satisfies even stronger piercing properties. We are able to show that this is best possible, as there are no (am+o(m))-piercing sequences for a<1/ln2. Our results allow for some new tight linear bounds for similar concepts defined for finite sequences.

The talk is based on the manuscript: https://arxiv.org/abs/2111.01887

We show that for no surface except for the plane does monadic second-order logic (MSO) have a zero-one-law – and not even a convergence law – on the class of (connected) graphs embeddable on the surface. In addition we show that every rational in [0,1] is the limiting probability of some MSO formula. This strongly refutes a conjecture by Heinig et al. (2014) who proved a convergence law for planar graphs, and a zero-one law for connected planar graphs, and also identified the so-called gaps of [0,1]: the subintervals that are not limiting probabilities of any MSO formula. The proof relies on a combination of methods from structural graph theory, especially large face-width embeddings of graphs on surfaces, analytic combinatorics, and finite model theory, and several parts of the proof may be of independent interest. In particular, we identify precisely the properties that make the zero-one law work on planar graphs but fail for every other surface.

Let's say m drivers have their favourite parking spot in the linear car park with n spots. Now, in some order, drivers will try to park their car at their favourite spot, and if they fail (because other car is standing there), they will try to park at the next avaible spot. If all drivers could park their car, we call this choices a parking function. In this seminar, we will look at this function proporties, create bijection from them to spanning forests and talk about some conjectures related to parking functions.

1. Richard Kenyon, Mei Yin. Parking functions: From combinatorics to probability. arXiv 2103.17180. (2021).
2. Wojciech Buczek. Parking functions. slides. (2021).

The paper provides a simple proof of a stronger version of Johansson-Molloy theorem, providing a bound on the list chromatic number of a graph based on maximum degree and neighbouhood density. The new proof only makes use of the first moment method. The counting argument used in the proof is inspired by work by Rosenfeld in the contex of non-repetitive graph coloring. The result is than extended to graphs with neighbourhoods with bounded density, which strengthens previous results. Lastly, the method is adapted to show asymptotically tight lowe bound on the number of colourings of sparse graphs .

1. François Pirot, Eoin Hurley. Colouring locally sparse graphs with the first moment method. arXiv 2109.15215. (2021).
2. Rafał Kilar. Colouring locally sparse graphs with the first moment method. slides. (2021).

In a recent breakthrough in scheduling, Batra, Garg, and Kumar gave the first constant approximation algorithm for minimizing the sum of weighted flow times. Wiese and I [STOC'21] managed to improve this large unspecified constant to (2+ϵ). I will give a very graphic presentation of the algorithmic techniques behind this.

A hat guessing game consists of a graph and bears assigned to vertices with a certain hat color. Each bear knows the colors of the bears belonging to the neighborhood of their vertex but does not know their own color. The bears win if at least one of them can guess the color of their hat. This presentation will introduce the aforementioned game, its variants and present findings of Václav Blažej, Pavel Dvořák and Michal Opler regarding fractional hat chromatic number of graphs with independence polynomials.

1. Václav Blažej, Pavel Dvořák, Michal Opler. Bears with Hats and Independence Polynomials. arXiv 2103.07401. 2021.
2. Marcin Serwin. Bears with Hats and Independence Polynomials. slides. (2021).

The paper introduces a new graph parameter called "weak degeneracy", a variant of the degeneracy parameter. The authors show various applications of weak degeneracy. For example, it turns out that this new parameter is strongly correlated with graph coloring. Authors derive alternative proofs for several classic upper bounds in graph coloring theory, including 5-list-coloring of planar graphs. My presentation will summarize the findings of the paper.

1. Anton Bernshteyn, Eugene Lee. Weak degeneracy of graphs. arXiv 2111.05908, (2021).
2. Krzysztof Potępa. Weak degeneracy of graphs. slides. (2021).

A graph G is semilinear of complexity t if the vertices of G are points in some real space, and the edges of G are determined by the sign-patterns of t linear functions. Many natural geometric graph families are semilinear of bounded complexity, e.g. intersection graphs of boxes, shift graphs, interval overlap graphs. There is a long line of research studying the exceptional Ramsey and coloring properties of such geometric graphs; I will show that many of these results can be traced back to their semilinear nature.

A class of fnite structures has a 0-1 law with respect to a logic if every property expressible in the logic has a probability approaching a limit of 0 or 1 as the structure size grows. To formulate 0-1 laws for maps (i.e., embeddings of graphs in a surface), it is necessary to represent maps as logical structures. Three such representations are given,
the most general being the full cross representation based on Tutte's theory of combinatorial maps. The main result says that if a class of maps has two properties, richness and large representativity, then the corresponding class of full cross representations has a 0-1 law with respect to first-order logic. As a corollary the following classes of maps
on a surface of fixed type have a first-order 0-1 law: all maps, smooth maps, 2-connected maps, 3-connected maps, triangular maps, 2-connected triangular maps, and 3-connected triangular maps

This paper begins with recounting known limits regarding Hadwiger's conjecture and related problems including list Hadwiger's conjecture, stating that there does exist constant c, such that K_t minor free graph G has list coloring number not exceeding ct. After the introduction, we are presented with proof that such constant has to be at least equal to 2, contrary to previous results, where c was bounded by 4/3 and conjectured to be equal to 3/2.

1. Krzysztof Michalik. Improved Lower Bound For The List Chromatic Number Of Graphs With No K_t Minor. slides. (2021).

Paper discusses properties of multivariate polynomials over finite grids, focaausing on he grids that are in some way "structured". To capture the degree to which a grid is structured, author introduces a notion of nullity, which can give us a numerical measure of structure. It is noted that for more structured grids we can obtain stronger versions of general theorems. This leads to the main results of the paper: the Structured Combinatorial Nullstellensatz and the Complete Coefficient Theorem.

1. Bogdan Nica. Polynomials over structured grids. arXiv 2110.05616. (2021).
2. Krzysztof Ziobro. Polynomials over structured grids. slides. (2021).

We elaborate the complexity of the Quantified Constraint Satisfaction Problem, QCSP(A), where A is a finite idempotent algebra. Such a problem is either in NP or is co-NP-hard, and the borderline is given precisely according to whether A enjoys the polynomially-generated powers (PGP) property. This completes the complexity classification problem for QCSPs modulo that co-NP-hard cases might have complexity rising up to Pspace-complete. Our result requires infinite languages, but in this realm represents the proof of a slightly weaker form of a conjecture for QCSP complexity made by Hubie Chen in 2012. The result relies heavily on the algebraic dichotomy between PGP and exponentially-generated powers (EGP), proved by Dmitriy Zhuk in 2015, married carefully to previous work of Chen. Finally, we discuss some recent work with Zhuk in which the aforementioned Chen Conjecture is actually shown to be false. Indeed, the complexity landscape for QCSP(B), where B is a finite constraint language, is much richer than was previously thought.

We consider the question of certifying unsatisfiability of random 3-CNF formulas. At which densities can we hope for a simple sufficient condition for unsatisfiability that holds almost surely? We study this question from the point of view of definability theory. The main result is that first-order logic cannot express any sufficient condition that holds almost surely on random 3-CNF formulas with $n^{2-\alpha}$ clauses, for any irrational positive number \alpha. In contrast, it can when the number of clauses is $n^{2+\alpha}$, for any positive \alpha. As an intermediate step, our proof exploits the planted distribution for 3-CNF formulas in a new technical way. Moreover, the proof requires us to extend the methods of Shelah and Spencer for proving the zero-one law for sparse random graphs to arbitrary relational languages.

Paper records open problems in computational geometry and related fields. For every problem, the authors provide a statement, origin, current status, partial results and related problems. My presentation focuses on a few chosen problems explained in a friendly manner.

1. Erik D. Demaine Joseph, S. B. Mitchell, and Joseph O’Rourke. The Open Problems Project. manuscript. (2020).
2. Artur Salawa. The Open Problems Project. slides. (2021).

A hypergraph is defined as a pair H = (V, E), where V is a set of vertices and E is a set of subsets of V - these subsets of vertices are called (hyper)edges. Graphs can be then seen as a concretization where all edges are sets of size 2. This can be shortly ascribed to the hypergraph as being 2-uniform. T-uniformity is a useful assumption for deriving its properties but sometimes one would wish for more general results. This approach is one of a few that are considered by the authors of the following paper which focuses on boundaries we can put on some characteristic properties of hypergraphs relating to their colorability and list-colorability. During the meeting, the basic concepts of hypergraphs and their colorability will be introduced and then the results of the paper will be interpreted alongside the presentation of the theorems and lemmas (and also an exemplar proof of one of them or two) which are used in the paper to attain the results.

1. Alexandr V. Kostochka, Douglas R. Woodall. Density Conditions for Panchromatic Colourings of Hypergraphs. Combinatorica 21, 515-541. (2001).
2. Grzegorz Wawrzesta. Density conditions for panchromatic colourings of hypergraphs - introduction and overview. slides. (2021).

W problemie Subset Sum, mając dany zbiór dodatnich liczb X oraz cel t, pytamy czy istnieje dowolny podzbiór sumujący się do dokładnie t. Należy on do klasy problemów NP-zupełnych, zatem naturalne jest badanie jego algorytmów aproksymacyjnych - takich, które szukają podzbioru sumującego się do co najmniej 1-ε wyniku optymalnego. Aktualnie najlepszy znany algorytm robi to w czasie O(min{n/ε,n+1/ε² log(1/ε)}). W szczególności nie jest znany żaden algorytm rozwiązujący ten problem w O((n+1/ε)^c) dla dowolnego c<2.

Autorzy w pracy pokazują równoważność tego problemu do Min-Plus-Convolution przeprowadzając redukcje w obie strony. Dzięki temu uzyskują algorytm aproksymacyjny dla Subset Sum o poprawionym czasie działania oraz udowadniają, że Subset Sum ma algorytm aproksymacyjny w czasie O((n+1/ε)^c) dla c<2 wtedy i tylko wtedy, gdy Min-Plus-Convolution może być rozwiązany w O(n^c') dla c'<2. Druga część równoważności jest jednak sprzeczna z hipotezą trudności tego problemu.

Dodatkowo, dla specjalnego wariantu Subset Sum zwanego Partition, autorzy stosują powyższą redukcję otrzymując algorytm aproksymacyjny działający w czasie O(n+1/ε^3/2). Jest to pierwszy deterministyczny algorytm z podkwadratową złożonością.

Circular-arc graphs, defined as the intersection graphs of arcs of a fixed circle, are probably one of the simplest classes of intersection graphs, which does not satisfy the Helly property in general (i.e. there are circular-arc graphs in which some cliques can be represented by arcs whose common intersection is empty). In particular, some cliques of a circular-arc G graph may satisfy the Helly property in some but not all circular-arc representations of G.

In the Helly Clique Problem, for a given circular-arc graph G and some of its cliques C₁,...,C_k (not necessarily maximal in G) one needs to decide whether there exists a circular-arc representation of G in which all the cliques C₁,...,C_k satisfy the Helly property. We know that the Helly Clique Problem is NP-complete and, under the ETH, it can not be solved in time 2^o(k)poly(n), where n is the number of vertices of G (Ağaoğlu et al.).

In the talk we will consider the Helly Clique Problem in the context of parameterized complexity, where the natural parameter is the number of cliques given in the input. We will show that the Helly Clique Problem:
- admits an algorithm with running time 2^{O(k log k)}poly(n) (and hence it belongs to FPT),
- admits a polynomial kernel of size O(k⁶).

Moreover, we will discuss the relation of the Helly Clique Problem with the recognition problems of so-called H-graphs; in particular, in the context of those graphs H for which the recognition problem remains open.

This is joint work with T. Krawczyk. The talk also includes joint work with D. Ağaoğlu, O. Cagrici, T. Hartmann, P. Hliněný, J. Kratochvíl, T. Krawczyk, and P. Zeman.

In this birdwatching paper our binoculars are focused upon a particular bird from Smullyan's enchanted forest of combinatory birds, to wit the Starling. In the feathers of lambda-calculus this bird has the plumage \abc:ac(bc). This term is usually named S, reminiscent of its inventor Schonfinkel and also the  combinatory ornithologist Smullyan. The combinator S is important for a variety of reasons. First, it is part of the S, K -basis for Combinatory Logic (CL). Second, there are several interesting questions and observations around S, mostly referring to termination and word problems. Our paper collects known facts, but poses in addition several new questions. For some of these we provide solutions, but several tough open questions remain.

In this paper authors propose approximation for Correlation Clustering problem with additional constaint of fairness. In a fair version of correlation clustering vertices have also colors and in the end in each cluster there should be "some" number of vertices of each color. What "some" means is dependent on variant of this constraint. Authors first reduce a problem of fair clustering correlation to a problem they call Fairlet Decomposition and then show approximation algorithm for this problem. In the end they describe some experiments they have done to prove Fair Correlation Clustering a viable version of Correlation Clustering.

1. Sara Ahmadian, Alessandro Epasto, Ravi Kumar, Mohammad Mahdian. Fair Correlation Clustering, arXiv 2002.02274. (2021).
2. Maciej Nemś. Fair Correlation Clustering. slides. (2021).

Paper surveys common part of two formal language issues. Issue of repetition free words and languages and issue of growth functions for words and languages. Paper gives an overview of current knowledge and search about an intersection of those two areas. It classifies power-free languages with regard to their growth rate. It also describes methods of esimating complexity of power-free languages paying attention to amount of computer resources needed by special method. Finally, it presents future directions of research in this area.

1. Karolina Gontarek. Growth properties of power-free languages. slides. (2021).

Autorzy analizują problem wypełniania krzyżówek, który już był rozważany na przykład przez Garey’a i Johnsona w ich książce „Computers and Intractability: A Guide to the Theory of NP-Completeness”. W problemie tym dostajemy m słów i n poziomych lub pionowych slotów (rubryk) oraz jesteśmy proszeni o wypełnienie ich tak, by przecięcia slotów się zgadzały. Autorzy próbują wskazać źródło trudności tej łamigłówki przyglądając się strukturze grafu przecięć slotów. Skupiają się na przypadku, gdy struktura tego grafu przypomina drzewo. Niestety, jeżeli przyjmiemy, że słowa nie mogą być używane wielokrotnie, okazuje się, że problem pozostaje NP-trudny nawet pod bardzo surowymi restrykcjami, jak na przykład, że graf przecięć jest sumą grafów typu star i alfabet ma rozmiar
2, lub że jest matchingiem (krzyżówka składa się z samych rozłącznych krzyży) z alfabetem rozmiaru 3. Zapytanie staje się prostsze, gdy pozwolimy na powtarzanie słów, gdyż autorom udaje się skonstruować algorytm działający w czasie m^tw, gdzie tw oznacza treewitdh (szerokość drzewiastą) grafu. Jednak nawet wtedy nie można algorytmu poprawić, by otrzymać FPT. Co więcej, pod założeniem ETH, problem nie może być rozwiązany w czasie m^o(k), gdzie k to liczba poziomych slotów instancji problemu.
Zmotywowani tymi przykrymi wynikami, autorzy rozważają bardziej ograniczony przypadek, gdzie parametryzujemy się liczbą slotów wszystkich, a nie tylko poziomych. Problem staje się FPT (o ile alfabet jest stałego rozmiaru), ale złożoność jest wykładnicza od n². Wykazują, że pod założeniem ETH, nie istnieje algorytm działający w czasie 2^{o(n^2)}, nawet dla dwuznakowego alfabetu. Pod koniec pracy rozważana jest wersja optymalizacyjna, w której staramy się wypełnić jak najwięcej pól. Łatwo jest wtedy uzyskać 1⁄2-aproksymację, nawet dla przypadku ważonego, wystarczy po prostu brać pod uwagę tylko pionowe albo tylko poziome sloty. Ten algorytm jest prawdopodobnie optymalny, gdyż istnienie lepszej aproksymacji działającej w wielomianowym czasie przeczyłoby Unique Games Conjecture. Ostatnie dwa wyniki są prawdziwe niezależnie, czy rozpatrujemy przypadek z powtarzaniem słów czy bez.

Local certification consists in assigning labels to the nodes of a graph in order to certify that some given property is satisfied, in such a way that the labels can be checked locally. In this talk, our goal is to certify that a graph G belongs to a given graph class. Such certifications exist for many sparse graph classes such as trees, planar graphs and graphs embedded on surfaces with labels of logarithmic size. It is open if such a certificate exist for any H-minor free graph class. We present some generic tools which allow us to certify the H-minor-free graphs (with logarithmic labels) for each small enough H.

More generally, we consider classes defined by any MSO formula (i.e. the MSO-model checking problem), and show a local version of the well-known Courcelle theorem: in bounded treedepth graphs, logarithmic certificates can be obtained for any MSO formula. We will also discuss many open problems related to  local certification of/on sparse graph classes.

Joint works with Laurent Feuilloley and Théo Pierron

A graph class C has asymptotic dimension at most d if for every r, the r-th distance power of any graph from C can be colored by d+1 colors so that every monchromatic connected subgraph has bounded weak diameter. In a recent breakthrough result, Bonamy, Bousquet, Esperet, Groenland, Liu, Pirot, and Scott proved that all proper minor-closed classes have asymptotic dimension at most two.  We investigate some questions motivated by this result for planar graphs and geometric intersection graphs.

Joint work with Sergey Norin

Authors in this work aim to establish various bounds and constraints on hypergraphs which are k-chromatic. Hypergraph is a graph where an edge don’t have to link exactly two vertices. Hypergraph is called simple, when none two of his edges has more then one common point, and is called clique when each two of his edges has at least one common point. Hyper graph is r-uniform when each of its edges contains exactly r points. Chromatic number is a smallest number k such that you can color points of the graph using k colors in the way that no edge is monochromatic. Main part of the work involves around the impact that being clique or simple has on 3-chromatic hypergraph structure. The main reason why those two things are connected is following trivial observation: If a hypergraph has chromatic number > 3 then it has two edges with exactly one common point.

1. Paul Erdős, László Lovász. Problems and results on 3-chromatic Hypergraphs and some related questions. Coll Math Soc J Bolyai. 10. (1974).
2. Roch Wójtowicz. Problems and results on 3-chromatic hypergraphs and some related questions. slides. (2021).

This paper explores almost proper graph colorings and list colorings - we allow the coloring to be improper, but we impose restrictions on the maximum number of neighbours of any vertex with the same color as the vertex itself (defect) or the maximum allowed size of a monochromatic connected graph component (clustering). The paper provides new bounds on coloring and list coloring number for sparse graphs, i.e. having bounded maximum average degree, taken over all subgraphs, or limited maximum degree. More precisely, the two main results of this paper are the new bounds on defective choosability and clustered choosability of graphs with bounded maximum average degree, being the best known results for graphs with unbounded maximum degree, but bounded maximum average degree, like k-stack and k-queue graphs.

1. Kevin Hendrey, David R. Wood. Defective and clustered choosability of sparse graphs. Combinatorics, Probability and Computing, 28, 791-810. (2019).
2. Grzegorz Gawryał. Defective and clustered choosability of sparse graphs. slides. (2021).

Stanislaw Ulam asked whether there exists a universal countable planar graph (that is, a countable planar graph that contains every countable planar graph as a subgraph). János Pach (1981) answered this question in the negative. We strengthen this result by showing that every countable graph that contains all countable planar graphs must contain an infinite complete graph as a minor. On the other hand, we construct a countable graph that contains all countable planar graphs and has several key properties such as linear colouring numbers, linear expansion, and every finite n-vertex subgraph has O(n^1/2) treewidth (which implies the Lipton-Tarjan separator theorem). More generally, for every fixed positive integer t we construct a countable graph that contains every countable K_t-minor-free graph and has the above key properties.

Joint work with Tony Huynh, Bojan Mohar, Robert Šámal and Carsten Thomassen

* exceptionally: Tuesday at 11:00

For a fixed graph H, let i(H, n) be the maximum induced density of H in any graph on n vertices. The limit i(H, n), as n goes to infinity, always exists and is called inducibility of H. Fox, Huang, and Lee proved that for almost all graphs H (think of large 'typical' graphs), inducibility of H can be obtained as the limit of induced density of H in its iterated blowups. Apart from that, inducibility is well understood only for narrow classes of graphs; in particular, it is still not determined for H being a path on 4 vertices.

In the reconfiguration problem, we are given a set of objects and rules of how one object can be reconfigured into another one. The main questions to be asked are if it is possible to reconfigure two given objects into each other (Reachability Problem) or how long is the shortest possible reconfiguration sequence. We focus on reconfiguring independent sets in a given graph. Two independent sets are reconfiguration-adjacent if their symmetric difference consists exactly of two vertices connected by an edge. It is known that for some graph classes the Reachability Problem can be solved in polynomial time. I briefly survey the topic and show that the problem is computationally hard for incomparability graphs. Moreover, I discuss the reconfiguration paths length problem in general and in more detail in the class of interval graphs.

1. Marcin Briański, Stefan Felsner, Jędrzej Hodor, Piotr Micek. Reconfiguring Independent Sets on Interval Graphs. arXiv, 2105.03402, (2021).
2. Marcin Brianski, Stefan Felsner, Jedrzej Hodor and Piotr Micek. Reconfiguring Independent Sets On Interval Graphs. slides. (2021).

In the early 1980s, Erdős and Sós, initiated the study of the classical Turán problem with a uniformity condition: the uniform Turán density of a hypergraph H is the infimum over all d for which any sufficiently large hypergraph with the property that all its linear-size subhyperghraphs have density at least d contains H. In particular, they raise the questions of determining the uniform Turán densities of K₄³, the complete 4-vertex 3-uniform hypergraph, and K₄³-, the hypergraph K₄³ with an edge removed. The latter question was solved only recently in [Israel J. Math. 211 (2016), 349–366] and [J. Eur. Math. Soc. 97 (2018), 77–97], while the former still remains open for almost 40 years.

Prior to our work, the hypergraph K₄³- was the only 3-uniform hypergraph with non-zero uniform Turán density determined exactly. During the talk, we will present the following two results:

• We construct a family of 3-uniform hypergraphs with uniform Turán density equal to 1/27. This answers a question of Reiher, Rödl and Schacht [J. London Math. Soc. 97 (2018), 77–97] on the existence of such hypergraphs, stemming from their result that the uniform Turán density of every 3-uniform hypergraph is either 0 or at least 1/27.
• We show that the uniform Turán density of the tight 3-uniform cycle of length k≥5 is equal to 4/27 if k is not divisible by three, and equal to zero otherwise. The case k=5 resolves a problem suggested by Reiher [European J. Combin. 88 (2020), 103117].

The talk is based on results obtained jointly with (subsets of) Matija Bucić, Jacob W. Cooper, Frederik Garbe, Ander Lamaison, Samuel Mohr and David Munhá Correia.

Several open problems related to 1-2-3 Conjecture are presented.

Matrix multiplication is one of the most basic linear algebraic operations outside elementary arithmetic. The study of matrix multiplication algorithms is very well motivated from practice, as the applications are plentiful. Matrix multiplication is also of great mathematical interest. Since Strassen's discovery in 1969 that n-by-n matrices can be multiplied asymptotically much faster than the brute-force O(n³) time algorithm, many fascinating techniques have been developed, incorporating ideas from computer science, combinatorics, and algebraic geometry.

The fastest algorithms over the last three decades have used Strassen's "laser method" and its optimization by Coppersmith and Winograd. The method has remained unchanged; the algorithms have differed in what the method was applied to. In recent work, joint with Josh Alman, we improve the method so that applying it to the same objects that the old method was applied to immediately yields faster algorithms. Using this new method, we obtain the theoretically fastest algorithm for matrix multiplication to date, with running time O(n^2.37286).

This talk will give an overview of the main techniques and will also outline our recent improvement of the laser method.

A word is repetitive if it contains two consecutive identical blocks. A sequence is non-repetitive up to mod r if each of its mod k (1⩽k⩽r) subsequences is non-repetitive. Let L be a language of non-repetitive (up to mod r) words. In this seminar, we are going to take a look at fixing blocks - a special kind of suffixes preventing words of L to have an extension in L. Using the fixing blocks method we are going to show some interesting properties of such languages. We also outline a method of attack for more complex problems.

1. James D. Currie, Cameron W. Pierce. The Fixing Block Method in Combinatorics on Words. Combinatorica 23, 571-584, (2003).
2. Szymon Salabura. The Fixing Block Method in Combinatorics on Words. slides. (2021).

(the seminar will only be online)

The age of an infinite word will be the set of all its finite subwords, it's essence will be the set of all finite subwords occurring infinitely many times. The language L_{121,323} is the language of all square-free infinite words, such that they don’t have 121 or 323 as subwords. It turns out if we consider the equivalence relations on L_{121,323} in respect to the ages and the essences we will get an uncountable cardinality of equivalence classes and 1 equivalence class respectively.

1. James D. Currie. Non-repetitive words: Ages and essences. Combinatorica 16, 19-40, (1996).
2. Wojciech Węgrzynek. Non-repetitive words: ages and essences. slides. (2021).

(the seminar will only be online)

In 1951, Kazimierz Zarankiewicz posed a problem asking for the largest possible number of edges in a bipartite graph that has a given number of vertices (n) and has no complete bipartite subgraphs of a given size. Although solved for some specific cases, it still remains open in general. It led to some interesting results in extremal graph theory, such as Kővári–Sós–Turán theorem which gives an upper bound for this problem. During the seminar, I will discuss several problems related to forbidding subgraphs, from easy up to unsolved ones. I will also show their connection with some geometric problems such as creating a maximum number of unit distances between n points on a plane.

1. Matoušek J. Incidence Problems. In: Matoušek J. (eds) Lectures on Discrete Geometry. Graduate Texts in Mathematics, vol 212. Springer, New York, NY. (2002).
2. Yufei Zhao. Graph Theory and Additive Combinatorics. course. MIT. (2020).
3. Bartosz Wodziński. Zarankiewicz's Problem and some related results. slides. (2021).

(the seminar will only be online)

A polyomino is a subset of R² formed by a strongly connected union of axis-aligned unit squares centered at locations on the square lattice Z². Let T = {T₁,T₂,...} be an infinite set of finite simply connected closed sets of R². Provided the elements of T have pairwise disjoint interiors and cover the Euclidean plane, then T is a tiling and the elements of T are called tiles. Provided every T_i ∈ T is congruent to a common shape T, then T is monohedral, T is the prototile of T, and the elements of T are called copies of T. In this case, T is said to have a tiling. We will go through some of the open problems related to polyomino tilings.

1. Michał Zwonek. Polyomino Tilings. slides. (2021).

(the seminar will only be online)

The notion of treewidth and tree decompositions plays a central role in the study of graphs with forbidden minors. Besides structural characterizations, the property of having boundedtreewidth, or a tree decomposision with certain "nice" properties, can be used algorithmically. However, until very recently, there were very few results that allowed to analyze the treewidth of graphs that exclude certain induced subgraphs. Indeed, a large clique has large treewidth, but is H-free for any graph H which is not a clique. It turns out that some intresting relations between the two worlds can be found if we additionally put some restrictions on vertex degrees - either just by bounding the maximum degree, or, in some cases, by bounding the degeneracy.

During the talk we will discuss some results of this flavor. In particular, we will show that

• graphs of bounded degeneracy that exclude all cycles of length at least t have bounded treewidth;
• graphs of bounded degree that exclude a fixed subdivision of the claw admit a certain type of an "*induced* grid/wall theorem": they either contain the line graph of a big subdivided wall as an *induced subgraph*, or have bounded treewidth.

Based on the joint work with Gartland, Lokshtanov, Pilipczuk, Pilipczuk, and with Abrishami, Chudnovsky, and Dibek.

The visibility graph of a polygon P is formed by the pairs of vertices u and v of P such that the segment uv is disjoint from the exterior of P. We show that the class of polygon visibility graphs is χ-bounded, thus answering a question by Kára, Pór, and Wood from 2005. Specifically, we prove the bound χ≤3⋅4^ω−1. We obtain the same bound for generalizations of polygon visibility graphs in which the polygon is replaced by a curve and straight-line segments are replaced by segments in a pseudo-line arrangement. The proof is carried through in the setting of ordered graphs. In particular, we show χ-boundedness of several classes of ordered graphs with excluded ordered substructures.

Joint work with James Davies, Tomasz Krawczyk, and Rose McCarty.

This is a part of Round the World Relay in Combinatorics organized by Oxford University.

Here is the full schedule: http://people.maths.ox.ac.uk/scott/relay.htm

And the zoom link for the whole event:

https://us02web.zoom.us/j/87593953555?pwd=UWl4eTVsanpEUHJDWFo3SWpNNWtxdz09

meeting id: 875 9395 3555
password: relay

Given a solution to a problem, we can try and apply a series of elementary operations to it, making sure to remain in the solution space at every step. What kind of solutions can we reach this way? How fast? This is motivated by a variety of applications, from statistical physics to real-life scenarios, including enumeration and sampling. In this talk, we will discuss various positive and negative results, in the special case of graph colouring.

For positive integers r<k<n let m(n,k,r) be the maximal size of a family F of k-element subsets of [n] such that no r vertices lie in more than one A in F. The Erdös-Hanani conjecture states that as n grows to infinity m(n,k,r) tends to (n choose r)/(k choose r). Firstly, we will see a sketch of the proof of this conjecture proposed by Vojtech Rödll. Then we will talk about how this is connected with packing in hypergraph and discuss the idea of an algorithm called Rödl nibble that achieves asymptotically optimal packing k-uniform hypergraphs.

1. Joonleng Tan. Rödl Nibble. manuscript. (2012).
2. Jan Mełech. Rödl Nibble. slides. (2021).

(the seminar will only be online)

Given a graph G, its (d,h)-decomposition is a partition of a set of edges of this graph into a d-degenerate graph and a graph with maximum degree at most h. We will study (d,h)-decomposability of planar graphs and consider the problem of finding minimum h_d such that every planar graph is (d,h_d)-decomposable. Since every planar graph is 5-degenerate, we will consider only the case of d less than 5.

1. Krzysztof Pióro. Decomposing planar graphs into graphs with degree restrictions. slides. (2021).

(the seminar will only be online)

We will overview the state-of-the-art results and techniques used in proofs of the lower bounds for constant depth circuits. We focus mostly on classes AC[0], ACC[0] and CC[0]. The most important challenges and some open problems are to be discussed.

Ant Colony Optimization algorithms are part of swarm intelligence approach to solving problems. They are inspired by behavior of ants. After finding a desired destination ants leave pheromones on the way back to the colony. This way all ants can detect the scent and decide to go in the direction suggested by pheromone trail. ACO is based on this concept and involves multi-agent computation. Communication between agents is done by changing the stimuli for all agents, to make a certain action. This is similar to ants leaving pheromones. Presentation will include basic concept of Ant Colony Optimization and an example of solving a well known problem using it. I will also present a formalization of ACO into a metaheuristic for combinatorial optimization problems. Presentation will end with talk about current state of ACO, its limitation and possible future.

1. M. Dorigo, M. Birattari, and T. Stutzle. Ant colony optimization. IEEE Computational Intelligence Magazine1(4), 28-39. (2006).
2. M. Dorigo and T. Stützle. Ant Colony Optimization: Overview and Recent Advances. In: Gendreau M., Potvin JY. (eds) Handbook of Metaheuristics. International Series in Operations Research & Management Science, vol 146. Springer, Boston, MA. (2010).
3. Maciej Nemś. Ant Colony Optimization. slides. (2021).
4. ANTS International Conference on Swarm Intelligence.

(the seminar will only be online)

Woodall’s conjecture tells us, that any directed cut with at least k edges has at least k disjoint dijoins. Set of edges D is a dijoin if and only if the digraph (V, E ∪ D^-1) is strongly connected. We will say about the linear programming formulation of this problem, equivalent and related problems to it, and some partial results by Shrijver, Lee and Wakabayashi, and Meszaros. We will also show counterexamples to a generalized version of the conjecture.

1. Paulo Feofiloff. Woodall’s conjecture on Packing Dijoins: a survey. manuscript. (2005)
2. Wojciech Buczek. Woodall’s conjecture. slides. (2021).

(the seminar will only be online)

In our paper [LICS'18] a generalization of Boolean circuits to arbitrary finite algebras was introduced and applied to sketch P versus NP-complete borderline for circuits satisfiability over algebras from congruence modular varieties. However nilpotent but not supernilpotent algebras have not been put on any side of this borderline.

This paper provides a broad class of examples, lying in this grey area, and show that, under the Exponential Time Hypothesis and Strong Exponential Size Hypothesis (saying that Boolean circuits need exponentially many modular counting gates to produce Boolean conjunctions of any arity), satisfiability over these algebras have intermediate complexity. We also sketch how these examples could be used as paradigms to fill the nilpotent versus supernilpotent gap in general.

Our examples are striking in view of the natural strong connections between circuits satisfiability and Constraint Satisfaction Problem for which the dichotomy was shown by Bulatov and Zhuk.

Joint work with Piotr Kawałek and Jacek Krzaczkowski

The discharging method is a technique that can be used to show that some global properties of a graph imply the existence of local structures. Then, we can sometimes show, that such structures imply that the graph has another global property. The discharging method is thus a "connector" between global properties of a graph (via local properties, e.g. having subgraphs or minors). In the first part of the presentation, we talk about the structure and coloring of sparse and plane graphs. Typical statements will sound like "If there is some global degree bound, then the chromatic number is somehow bounded"

1. D. Cranston, D. West. A Guide to the Discharging Method. manuscript. (2013).
2. Vladyslav Rachek, Ruslan Yevdokymov. An Introduction to the Discharging Method via Graph Coloring. slides. (2021).

(the seminar will only be online)

Given an n-by-n generator matrix with entries being subsets of a fixed field we generate the set of all matrices with entries coming from the corresponding entries in the generator matrix. Such a set of matrices is strongly singular if each member is a singular matrix. In this talk I will survey natural generalizations and connections to other problems. In particular, I will describe algorithm by Geelen for  maximum rank matrix completion problem.

The conjecture deals with queries on graph. More concretely given property of a graph (such as connectedness or non-emptiness) we may ask whether it is possible to recognize a graph with this property without querying all of its edges. It turns out that for many properties it is indeed not possible to do so in a deterministic manner for all graphs. The Aanderaa–Karp–Rosenberg conjecture states that any non-trivial monotone graph property cannot be determined by a deterministic algorithm with less than n(n-1)/2 queries. Such graph properties are called evasive, thus this conjecture is sometimes called evasiveness conjecture.

1. Marcin Serwin. Aanderaa-Karp-Rosenberg conjecture. slides. (2021).

(the seminar will only be online)

In the edge orientation problem, one is asked to orient edges of a given graph such that the out-degrees of vertices are bounded by some function. In the fully dynamic variant, we want to process arbitrary edge insertions and deletions in an online fashion. We will show an algorithm for graphs with bounded arboricity that achieves logarithmic out-degree bound and supports updates in O(log n) worst-case time.

1. Tsvi Kopelowitz, Robert Krauthgamer, Ely Porat, Shay Solomon. Orienting Fully Dynamic Graphs with Worst-Case Time Bounds. ICALP 2014, LNCS 8573, 532-543, (2014). (arXiv:1312.1382)
2. Krzysztof Potępa. Orienting Fully Dynamic Graphs with Worst-Case Time Bounds. slides. (2021).

(the seminar will only be online)

Given a graph class C, a graph G is universal for C if it "contains" all the graphs from C. As there are several notions of containment, there are several notions of universal graphs. In this talk I'll mention two versions:

• induced-universal graphs: here G contains all the graphs of C as induced subgraphs
• isometric-universal graphs: here G contains isometric copies of all the graphs from C (isometric means that the distances are preserved in the embedding)

Note that an isometric copy is an induced copy, so the second notion is stronger. These notions are closely related to the notion of labelling schemes in graphs. The goal is to assign labels to the vertices of each graph G from C such that upon reading the labels of any two vertices u and v, we know some properties of u and v in G (whether they are adjacent, or their distance, or whether u is reachable from v if G is a digraph). It turns out that minimizing the size of the labels is equivalent to constructing small universal graphs, at least in the case of induced-universal graphs. For isometric-universal graphs some additional work needs to be done.

I'll survey some recent progress in this area. In particular I'll show how to construct induced-universal graphs of almost optimal size for any hereditary class, using the regularity lemma. In particular this implies almost optimal reachabilty labelling schemes in digraphs and comparability labelling schemes in posets, and the construction of an almost optimal universal poset for the class of all n-element posets (of size 2^n/4+o(n)). I will also show how to construct isometric-universal graphs of size 3^n+o(n) for the class of all n-vertex graphs, answering a question of Peter Winkler.

Based on joint work with Marthe Bonamy, Cyril Gavoille, Carla Groenland, and Alex Scott.

There is a close connection between minors of the graph and a lower bound on such number k that each edge (or vertex) belongs to at least k triangles. One of the most interesting classes of minors is the class of complete graphs K_r. In the paper 'On triangles in K_r-minor free graphs', Boris Albar and Daniel Gonçalves take a closer look at this class of graphs. Based on their work I will present some interesting results regarding this connection and show how it correlates with Hadwiger's conjecture.

1. Boris Albar Daniel Gonçalves. On triangles in K_r‐minor free graphs. Journal of Graph Theory, 88(1), 154-173, (2017). (arXiv:1304.5468)
2. Mateusz Kaczmarek. On triangles and minors. slides. (2021).

(the seminar will only be online)

A combinatorial structure is said to be quasirandom if it resembles a random structure in a certain robust sense. The notion of quasirandom graphs, developed in the work of Rödl, Thomason, Chung, Graham and Wilson in 1980s, is particularly robust as several different properties of truly random graphs, e.g., subgraph density, edge distribution and spectral properties, are satisfied by a large graph if and only if one of them is.

We will discuss quasirandom properties of other combinatorial structures, tournaments, permutations and Latin squares in particular, and present several recent results obtained using analytic tools of the theory of combinatorial limits.

The talk is based on results obtained with different groups of collaborators, including Timothy F. N. Chan, Jacob W. Cooper, Robert Hancock, Adam Kabela, Ander Lamaison, Taísa Martins, Roberto Parente, Samuel Mohr, Jonathan A. Noel, Yanitsa Pehova, Oleg Pikhurko, Maryam Sharifzadeh, Fiona Skerman and Jan Volec.

An acyclic vertex coloring of a graph is a proper vertex coloring such that there are no bichromatic cycles. The acyclic chromatic number of G, denoted as a(G), is the minimum number of colors required for acyclic vertex coloring of graph G. Known problem in this area is to find an upper bound for an acyclic chromatic number for graphs with a fixed maximum degree. One of the first papers on this topic is Hocquard's article "Graphs with maximum degree 6 are acyclically 11-colorable". The proofing technique from his work has been used in many later papers that show similar bounds for graphs with fixed maximum grades.

1. Hervé Hocquard. Graphs with maximum degree 6 are acyclically 11-colorable. Information Processing Letters, 111(15), 748-753, (2011). (manuscript)
2. Bartłomiej Jachowicz. Acyclic coloring of graphs with fixed maximum degree. slides. (2021).

(the seminar will only be online)

We revisit the problem of maintaining the longest increasing subsequence (LIS) of an array under

(i) inserting an element, and

(ii) deleting an element of an array.

In a recent breakthrough, Mitzenmacher and Seddighin [STOC 2020] designed an algorithm that maintains an O((1/ϵ)^O(1/ϵ))-approximation of LIS under both operations with worst-case update time ~O(n^ϵ), for any constant ϵ>0. We exponentially improve on their result by designing an algorithm that maintains a (1+ϵ)-approximation of LIS under both operations with worst-case update time ~O(ϵ⁻⁵). Instead of working with the grid packing technique introduced by Mitzenmacher and Seddighin, we take a different approach building on a new tool that might be of independent interest: LIS sparsification.

While this essentially settles the complexity of the approximate version of the problem, the exact version seems more elusive. The only known sublinear solution was given very recently by Kociumaka and Seddighin [STOC 2021] and takes ~O(n^2/3) time per update. We show polynomial conditional lower bounds for two natural extensions of this problem: weighted LIS and LIS in any subarray.

Joint work with Wojciech Janczewski

The Hadwiger Conjecture states that if a graph G does not contain a clique on t vertices as a minor, then G is (t-1)-colorable. For low values of t (<7) it was already shown that this claim is actually true. Currently, the best-known upper bound on the chromatic number of K_t-minor-free graphs is O(ct*sqrt(log(t))) and the proof follows from a degeneracy argument. Unfortunately, this approach cannot be exploited further. However, by revisiting Thomassen's reasoning from '94 we can try to prepare the ground for a new way of attacking the Hadwiger Conjecture based on graph choosability. For that, we will take a look at a new proof of a theorem that every K₅-minor-free graph is 5-choosable.

1. David R. Wood, Svante Linusson. Thomassen's Choosability Argument Revisited. SIAM Journal on Discrete Mathematics, 24(4), 1632-1637, (2010). (arXiv:1005.5194)
2. Piotr Mikołajczyk. Thomassen's choosability argument revisited. slides. (2021).

(the seminar will only be online)

In 2015, Felsner, Trotter, and Wiechert showed that posets with outerplanar cover graphs have bounded dimension. We generalise this result to posets with k-outerplanar cover graphs. Namely, we show that posets with k-outerplanar cover graph have dimension O(k³). As a consequence, we show that every poset with a planar cover graph and height h has dimension O(h³). This improves the previously best known bound of O(h⁶) by Kozik, Micek and Trotter.

Joint work with Maximilian Gorsky

https://arxiv.org/abs/2103.15920

Proposed by Noga Alon in 1999 an algebraic technique inspired by Hilbert’s Nullstellensatz. Despite being an observation about roots of a polynomial in n variables, it finds a usage in numerous fields - from Combinatorial Number Theory to Graph Theory. The theory itself is simple, but can be used in ingenious ways - appearing even as almost a necessary step to solve a problem during the 2007 IMO competition. During this time slot I will present a theorem and prove it with as I believe a simpler proof constructed by Mateusz Michałek, that is using a basic induction idea over a polynomial degree. Finally we will again follow the original N. Alon paper to see applications of a theorem in the graph coloring problems and some more.

1. Noga Alon. Combinatorial Nullstellensatz. Combinatorics Probability and Computing, 8 (1999), 7-29. (manuscript)
2. Mateusz Michałek. A short proof of Combinatorial Nullstellensatz. American Mathematical Monthly, 117 (2010), 821-823. (arXiv:0904.4573)
3. Tomasz Kochanek. Combinatorial Nullstellensatz. (2012) (pl). manuscript.
4. Jędrzej Kula. Combinatorial Nullstellensatz. slides. (2021).

(the seminar will only be online)

We present a progress on local computation algorithms for two coloring of k-uniform hypergraphs. We focus on instances that (for a parameter α) satisfy strengthened assumption of Local Lemma of the form 2^1-αk(Δ+1)e<1, where Δ is the bound on the maximum edge degree of the hypergraph. We discuss how previous works on the subject can be used to obtain an algorithm that works in polylogarithmic time per query for α up to about 0.139. Then, we present a procedure that, within similar bounds on running time, solves wider range of instances by allowing α to be at most about 0.227.

Joint work with Jakub Kozik

In 1981, Kelly showed that planar posets can have arbitrarily large dimension. However, the posets in Kelly's example have bounded Boolean dimension and bounded local dimension, leading naturally to the questions as to whether either Boolean dimension or local dimension is bounded for the class of planar posets. The question for Boolean dimension was first posed by Nešetril and Pudlák in 1989 and remains unanswered today. The concept of local dimension is quite new, introduced in 2016 by Ueckerdt. In just the last year, researchers have obtained many interesting results concerning Boolean dimension and local dimension, contrasting these parameters with the classic Dushnik-Miller concept of dimension, and establishing links between both parameters and structural graph theory, path-width and tree-width in particular. Here we show that local dimension is not bounded on the class of planar posets. Our proof also shows that the local dimension of a poset is not bounded in terms of the maximum local dimension of its blocks, and it provides an alternative proof of the fact that the local dimension of a poset cannot be bounded in terms of the tree-width of its cover graph, independent of its height. This is a joint work with Jarosław Grytczuk and W.T. Trotter.

Bartłomiej Bosek, Jarosław Grytczuk, William T. Trotter. Local Dimension is Unbounded for Planar Posets. arXiv, pages 1-12, 2017.

(the seminar will only be online)

Graph classes excluding a path or a subdivided claw as an induced subgraph are so far mysterious: on one hand, beside some corner cases, they do not seem to have any good structural description, but on the other hand, a number of combinatorial problems - including Maximum Independent Set (MIS) - lack an NP-hardness proof in these graph classes, indicating a possible hidden structure that can be used algorithmically. Furthermore, graphs excluding a fixed path as an induced subgraph were one of the earliest examples of a chi-bounded graph class, with an elegant proof technique dubbed the Gyarfas' path argument. In the recent years the progress accelerated, leading to, among other results, (a) a quasi-polynomial-time algorithm for MIS in graphs excluding a fixed path as an induced subgraph, (b) a QPTAS for MIS in graphs excluding a subdivided claw as an induced subgraph, (c) the proof of the Erdos-Hajnal property in graph classes excluding a fixed forest and its complement. In the talk, I will survey these results, showing the role of the Gyarfas' path argument in most (all?) of them, and outline research directions for the future.

A given pair of two incomparable elements x, y in poset P is called balanced if, of all line extensions P, the element x lies above y by at most 2/3 and on at least 1/3 of all extensions of the poset P. The 1/3 - 2/3 conjecture says that any poset that is not linear has a balanced pair. The talk presents basic definitions and an overview of the most important results in this field.

(the seminar will only be online)

In this talk we survey results for pseudocircle arrangements. Along the way we present several open problems. Among others we plan to touch the following topics:

 * The taxonomy of classes of pseudocircle arrangements.
 * The facial structure with emphasis on triangles and digons.
 * Circularizability.
 * Coloring problems for pseudocircle arrangements.

The talk includes work of Grünbaum, Snoeyink, Pinchasi, Scheucher, myself, and others.

Centered graph coloring is graph coloring, such that for every connected subgraph, this subgraph contains a vertex with a unique color (we call such a vertex center). Linear coloring is coloring, such that every path has a center. We denote by cen(G) and lin(G) respectively, a minimal number of colors needed. It is obvious that lin(G) ≤ cen(G). What about the other direction? Authors of the paper show that cen ≤ f(lin), where f is a polynomial of the degree 190. Moreover, the authors conjecture that cen ≤ 2 lin for every graph. What is interesting, we don't know how to prove such abound even for trees. Luckily, for some classes of graphs, we can do better than 190-poly. During the seminar, I will present results for simple classes of graphs and I will try to sketch the general proof. In particular, I will present a cubic bound for interval graphs. The proof in the paper is incorrect, but I and dr Micek managed to fix it. Finally, I will present our new result for graphs with bounded path width.

1. Jeremy Kun, Michael P. O’Brien, Marcin Pilipczuk, and Blair D. Sullivan. Polynomial Treedepth Bounds in Linear Colorings. Algorithmica. volume 83, pages 361–386. 2021.
2. Jędrzej Hodor. Polynomial Treedepth Bounds in Linear Colorings. slides. 2021.

(the seminar will only be online)

We describe a polynomial-time algorithm which, given a graph G with treewidth t, approximates the pathwidth of G to within a ratio of O(t √ log t). This is the first algorithm to achieve an f(t)-approximation for some function f.

Our approach builds on the following key insight: every graph with large pathwidth has large treewidth or contains a subdivision of a large complete binary tree. Specifically, we show that every graph with pathwidth at least th+2 has treewidth at least t or contains a subdivision of a complete binary tree of height h+1. The bound th+2 is best possible up to a multiplicative constant. This result was motivated by, and implies (with c=2), the following conjecture of Kawarabayashi and Rossman (SODA'18): there exists a universal constant c such that every graph with pathwidth Ω(k^c) has treewidth at least k or contains a subdivision of a complete binary tree of height k.

Our main technical algorithm takes a graph G and some (not necessarily optimal) tree decomposition of G of width t' in the input, and it computes in polynomial time an integer h, a certificate that G has pathwidth at least h, and a path decomposition of G of width at most (t'+1)h+1. The certificate is closely related to (and implies) the existence of a subdivision of a complete binary tree of height h. The approximation algorithm for pathwidth is then obtained by combining this algorithm with the approximation algorithm of Feige, Hajiaghayi, and Lee (STOC'05) for treewidth.

Joint work with Carla Groenland, Gwenaël Joret, and Wojciech Nadara.

One of the variations of coloring a graph is assigning colors to vertices such that for each vertex v, at most half of the neighbors of v have the same color as v. Such coloring is called majority coloring of the graph. The goal is to color the graph as a majority with the fewest possible colors. During the presentation, various variants of this problem, historical results, the latest results as well as still intriguing hypotheses will be discussed. Among other things, the results of joint cooperation with Marcin Anholcer, Jarosław Grytczuk, and Gabriel Jakóbczak will be presented.

1. László Lovász, On decomposition of graphs Studia Scientiarum Mathematicarum Hungarica. A Quarterly of the Hungarian Academy of Sciences, 1, 237–238, 1966.
2. Paul D. Seymour, On the two-colouring of hypergraphs, The Quarterly Journal of Mathematics. Oxford. Second Series, 25, 303–312, 1974.
3. Dominic van der Zypen, Majority coloring for directed graphs MathOverflow, 2016.
4. Stephan Kreutzer, Sang-il Oum, Paul D. Seymour, Dominic van der Zypen, and David R. Wood, Majority colourings of digraphs, Electronic Journal of Combinatorics, 24(2):Paper 2.25, 9, 2017.
5. Marcin Anholcer, Bartłomiej Bosek, Jarosław Grytczuk, Majority Choosability of Digraphs Electronic Journal of Combinatorics, 24(3), Paper 3.57, 2017.
6. António Girao, Teeradej Kittipassorn, Kamil Popielarz, Generalized majority colourings of digraphs, Combinatorics, Probability and Computing, 26(6), 850–855, 2017.
7. Fiachra Knox and Robert Samal, Linear bound for majority colourings of digraphs, Electronic Journal of Combinatorics, 25(3):Paper 3.29, 4, 2018.
8. Bartłomiej Bosek, Jarosław Grytczuk, Gabriel Jakóbczak, Majority Coloring Game, Discrete Applied Mathematics, 255, 15–20, 2019.

(the seminar will only be online)

We study the problem of learning a linear model to set the reserve price in an auction, given contextual information, in order to maximize expected revenue from the seller side. First, we show that it is not possible to solve this problem in polynomial time unless the Exponential Time Hypothesis fails. Second, we present a strong mixed-integer programming (MIP) formulation for this problem, which is capable of exactly modeling the nonconvex and discontinuous expected reward function. More broadly, we believe this work offers an indication of the strength of optimization methodologies like MIP to exactly model intrinsic discontinuities in machine learning problems. This presentation might be of interest for, among the others, the participants of the Algorithmic Game Theory course as it presents the modern approach for maximizing revenue in second-price auctions.

1. Joey Huchette, Haihao Lu, Hossein Esfandiari, Vahab Mirrokni. Contextual Reserve Price Optimization in Auctions via Mixed-Integer Programming. arXiv:2002.08841. 2020.
2. Kamil Kropiewnicki. Contextual Reserve Price Optimization in Auctions via Mixed-Integer Programming. slides. 2021.

(the seminar will only be online)

Let G₁, G₂ be n-vertex graphs. We say that they pack if they are edge-disjoint subgraphs of a complete graph on n vertices. The Bollobás-Eldridge-Catlin conjecture states that if (Δ(G₁) + 1) (Δ(G₂) + 1) < n + 1, then G₁ and G₂ pack. During the seminar, we will take a look at current results related to this problem, i.e. classes of graphs for which it has been proven as well as similar conjectures stemming from it.

1. Rafał Burczyński. Bollobás-Eldridge-Catlin Conjecture. slides. 2021.

(the seminar will only be online)

The cap set problem asks how large can a subset of Z_/3Zⁿ be and contain no lines or, more generally, how can large a subset of Z_/pZⁿ be and contain no arithmetic progression. The problem was open until 2016 when its basic version was solved. During the lecture, we'll see the motivation for thinking about this. It appears there are some interesting applications of this result in combinatorics, geometry, and even board games.

1. Weronika Lorenczyk. Cap Conjecture - motywacje i zastosowania. slides. 2021. (Polish)

(the seminar will only be online)

Let H be a graph on h vertices. The Graph Removal Lemma states that for any ε > 0, there exists a constant δ(ε, H) > 0 such that for any n-vertex graph G with fewer than δn^h subgraphs isomorphic to H, it is possible to eliminate all copies of H by removing at most εn² edges from G. It has several important consequences in number theory, discrete geometry, graph theory, and computer science.

During the seminar, I will discuss this lemma and its extensions. I will also tell about some of its applications, such as graph property testing and Szeremedi's Theorem proof.

1. David Conlon, Jacob Fox. Graph removal lemmas. arXiv:1211.3487. 2012.
2. Bartosz Wodziński. Graph removal lemma. slides. 2021.

(the seminar will only be online)

Machine learning has already leaked almost all areas. What about Combinatorial Optimization? At this seminar, I will present basic ML concepts and methods in CO: Where you can add ML black box in your algorithm? Can heuristics be compared to ML? What are the recent achievements?

1. Artur Kasymov. Machine learning in Combinatorial Optimization. slides. 2021.

(the seminar will only be online)

Consider the following absurd argument concerning planar, binary, rooted, unlabelled trees. Every such tree is either the trivial tree or consists of a pair of trees joined together at the root, so the set T of trees is isomorphic to 1+T². Pretend that T is a complex number and solve the quadratic T = 1+T² to find that T is a primitive sixth root of unity and so T⁶ = 1. Deduce that T⁶ is a one-element set; realize immediately that this is wrong. Notice that T⁷ = T is, however, not obviously wrong, and conclude that it is therefore right. In other words, conclude that there is a bijection T⁷ ≅ T built up out of copies of the original bijection T ≅ 1+T²: a tree is the same as seven trees.

1. Andreas Blass. Seven Trees in One. arXiv:math/9405205. 1994.
2. Marcelo Fiore, Tom Leinster. Objects of Categories as Complex Numbers. arXiv:math/0212377. 2002.
3. Bruno Pitrus. Seven trees in one: objects of categories as complex numbers. slides. 2021.

(the seminar will only be online)

A path decomposition of a graph G is a collection of edge-disjoint paths of G that covers the edge set of G. Gallai (1968) conjectured that every connected graph on n vertices admits a path decomposition of cardinality at most ⌈n/2⌉. Gallai’s Conjecture has been verified for many classes of graphs. In this seminar, we will cover some of these graph classes.

1. Krzysztof Pióro. Hipoteza Gallai. slides. 2020.

(the seminar will only be online)

In this seminar, I will cover general ideas that stand behind Aleph protocol. Aleph is a leaderless, fully asynchronous, Byzantine fault tolerant consensus protocol for ordering messages exchanged among processes. It is based on a distributed construction of a partially ordered set and the algorithm for reaching a consensus on its extension to a total order.

1. Adam Gągol, Michał Świetek. Aleph: A Leaderless, Asynchronous, Byzantine Fault Tolerant Consensus Protocol. arXiv:1810.05256. 2018.
2. Adam Gągol, Damian Leśniak, Damian Straszak, Michał Świetek. Aleph: Efficient Atomic Broadcast in Asynchronous Networks with Byzantine Nodes. arXiv:1908.05156. 2019.
3. Adam Gągol, Damian Straszak. Blockchain Fundamentals. 2020.
4. Szymon Żak. Aleph: Efficient Atomic Broadcast in Asynchronous Networks with Byzantine Nodes. slides. 2021.

(the seminar will only be online)

Graph is vertex-transitive if every vertex has the same local environment, so that no vertex can be distinguished from any other based on the vertices and edges surrounding it. In 1969, Lovasz conjectured that every finite connected vertex-transitive graph has Hamiltonian path. Moreover, up to now there are currently only five known connected vertex-transitive graphs not containing Hamiltonian cycle. In this seminar we will focus also on some other weaker variants of Lovasz conjecture related to other interesting class of graphs that encode the abstract structures of a groups - Cayley graphs.

1. Jan Mełech. Hamiltonian paths/cycles in vertex-transitive/symmetric graphs. slides. 2020.

(the seminar will only be online)

Recent work on the combinatorics of the linear lambda term shows that it has various connections to the theory of graph surfaces (maps). Based on paper [1] I will present a bijection between linear lambda terms (presented as diagrams) and rooted trivalent maps. Also, I will cover the recent conjecture proposed in 2019 that a special class of planar lambda terms can be counted the same way that rooted bicubic maps.

1. Noam Zeilberger. Linear lambda terms as invariants of rooted trivalent maps. arXiv:1512.06751. 2015.
2. Mateusz Kaczmarek. From linear lambda terms to rooted trivalent maps. slides. 2020.

(the seminar will only be online)

Let C be a Jordan curve. We say that polygon P is inscribed in C if all vertices of P belong to C. In the inscribed square problem we ask if every Jordan curve admits an inscribed square. It's also known as "Toeplitz’s conjecture" or the "Square peg problem". In this seminar, we will show some equivalent problems to this conjecture and focus on special cases of the Jordan curves.

1. Wojciech Buczek. Inscribed square problem. slides. 2020.

(the seminar will only be online)

The Hamiltonian Cycle problem is one of the oldest and most common NP-complete problems. It consists of checking whether in a given graph there is a cycle visiting each vertex exactly once. I will present a parameterized algorithm based on graph tree decomposition. Assuming that a nice tree decomposition of the width k is known at the input Hamiltonian cycle problem can be solved in a time 2^(O(k))n^(O(1)).

1. Bartłomiej Jachowicz. Parameterized by treewidth algorithms for Hamiltonian Cycle. slides. 2020.

(the seminar will only be online)

Given a finite metric space (V,d), an approximate distance oracle is a data structure which, when queried on two points u,v∈V, returns an approximation to the actual distance between u and v which is within some bounded stretch factor of the true distance. The first work in this area was done by Mikkel Thorup and Uri Zwick, they devised an oracle with construction time being O(kmn^(1/k)) and with the space complexity of O(kn^(1+1/k)). The achieved stretch, that is the measure of how accurate the answer by the approximate oracle will be, is bounded by (2k-1). The query time is O(k), this has been subsequently improved to O(log n) by Wulff-Nilsen and to O(1) by Shiri Chechik.

1. Michał Zwonek. Approximate Distance Oracles. slides. 2020.

(the seminar will only be online)

A connected graph G is called double-critical if the chromatic number of G decreases by two if any two adjacent vertices of G are removed. In 1966, Erdős and Lovász conjectured that the only double-critical n-chromatic graph is the complete graph on n vertices. During the seminar, I will present what has been verified about the conjecture.

1. Wojciech Grabis. Double-critical graph conjecture. slides. 2020.

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A well-known theorem of Erdős states that there exists a graph on n vertices, with no clique or independent set of a size larger than O(log n). The Erdős–Hajnal conjecture says it is very different if we consider families of graphs defined by forbidden induced subgraphs. Specifically, it is conjectured that for every graph H, there exists a constant δ(H) such that every H-free graph G has either a clique or independent set of size |V(G)|^δ(H). We will discuss some classes of graphs for which the conjecture has been proven, as well as weaker theorems that hold for all graphs.

1. Krzystzof Potępa. The Erdős–Hajnal conjecture. slides. 2020.

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An (m,n)-cycle cover is a covering of a graph consisting of m cycles such that every edge is covered exactly n times. For positive integers m, n it is natural to ask what family of graphs have (m,n)-cycle covers. The answers are known for some values, but for many others, they are conjectured or totally open.

1. Marcin Serwin. (m,n)-cycle cover conjectures. slides. 2020.

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A harmonious coloring of a k-uniform hypergraph H is a vertex coloring such that no two vertices in the same edge share the same color, and each k-element subset of colors appears on at most one edge. The harmonious number h(H) is the least number of colors needed for such a coloring. We prove that k-uniform hypergraphs of bounded maximum degree Δ satisfy h(H) = O(k√k!m), where m is the number of edges in H which is best possible up to a multiplicative constant. Moreover, for every fixed Δ, this constant tends to 1 with k → ∞. We use a novel method, called entropy compression, that emerged from the algorithmic version of the Lovász Local Lemma due to Moser and Tardos. This is joint work with Sebastian Czerwinski, Jarosław Grytczuk, and Paweł Rzazewski.

1. Bartłomiej Bosek, Jarosław Grytczuk, Paweł Rzążewski, Sebastian Czerwiński. Harmonious coloring of uniform hypergraphs. Applicable Analysis and Discrete Mathematics, 10(1), 73-87, 2016.

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A few years ago M. Might et al. published somehow unusual approach to parsing context-free grammars by using derivative operator. Later it was proven, that its worst case complexity is polynomial, putting it on a par with other classical approaches. We introduce an elegant generalization to this method by a generic algorithm parametrized with a semiring. Depending on the chosen algebra we can obtain polynomial algorithms for problems like parsing, recognizing or counting parse trees for CFGs.

1. Piotr Mikołajczyk. Polynomial algorithms for CFGs via semiring embedding. slides. 2020.

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One of the variations of coloring a graph is assigning colors to vertices such that for each vertex v, at most half of the neighbors of v have the same color as v. Such coloring is called majority coloring of the graph. The goal is to color the graph as a majority with the fewest possible colors. During the presentation, various variants of this problem, historical results, the latest results as well as still intriguing hypotheses will be discussed. Among other things, the results of joint cooperation with Marcin Anholcer, Jarosław Grytczuk, and Gabriel Jakóbczak will be presented.

1. László Lovász, On decomposition of graphs Studia Scientiarum Mathematicarum Hungarica. A Quarterly of the Hungarian Academy of Sciences, 1, 237–238, 1966.
2. Paul D. Seymour, On the two-colouring of hypergraphs, The Quarterly Journal of Mathematics. Oxford. Second Series, 25, 303–312, 1974.
3. Dominic van der Zypen, Majority coloring for directed graphs MathOverflow, 2016.
4. Stephan Kreutzer, Sang-il Oum, Paul D. Seymour, Dominic van der Zypen, and David R. Wood, Majority colourings of digraphs, Electronic Journal of Combinatorics, 24(2):Paper 2.25, 9, 2017.
5. Marcin Anholcer, Bartłomiej Bosek, Jarosław Grytczuk, Majority Choosability of Digraphs Electronic Journal of Combinatorics, 24(3), Paper 3.57, 2017.
6. António Girao, Teeradej Kittipassorn, Kamil Popielarz, Generalized majority colourings of digraphs, Combinatorics, Probability and Computing, 26(6), 850–855, 2017.
7. Fiachra Knox and Robert Samal, Linear bound for majority colourings of digraphs, Electronic Journal of Combinatorics, 25(3):Paper 3.29, 4, 2018.
8. Bartłomiej Bosek, Jarosław Grytczuk, Gabriel Jakóbczak, Majority Coloring Game, Discrete Applied Mathematics, 255, 15–20, 2019.

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A given pair of two incomparable elements x, y in poset P is called balanced if, of all line extensions P, the element x lies above y by at most 2/3 and on at least 1/3 of all extensions of the poset P. The 1/3 - 2/3 conjecture says that any poset that is not linear has a balanced pair. The talk presents basic definitions and an overview of the most important results in this field.

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Let P be a set of n points in R², ε > 0. A set of points Q is called a weak ε-net for P with respect to a family S of objects (e.g. axis-parallel rectangles or convex sets) if every set from S containing more than εn points of P contains a point from Q. Let R be the family of all axis-parallel rectangles in R² and ε^R_k be the smallest real number such that for any P there exists a weak ε^R_k-net of size k. The work by Aronov et al. suggests that the inequality ε^R_k ≤ 2/(k+3) may hold. In this talk we present the complete proofs of this inequality for k=1,...,5 and prove that this bound is tight for k=1,2,3. Besides, it is not clear how to construct optimal nets. Langerman conjectured that k-point 2/(k+3)-nets can be chosen from some speciffc set of points which are the intersections of grid lines, where the grid is of size k×k. We give counterexamples to this conjecture for nets of size 3 through 6.

1. Vladyslav Rachek. Small weak epsilon-nets in families of rectangles. Bachelor's Thesis. Jagiellonian University. 2020.
2. B. Aronov, F. Aurenhammer, F. Hurtado, S. Langerman, D. Rappaport, C.Seara, and S. Smorodinsky. Small weak epsilon-nets. Computational Geometry: Theory and applications. 42(5), 2009. pp. 455-462.
3. Vladyslav Rachek. On small weak epsilon-nets for axis-parallel rectangles. slides. 2020.

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A series of open (and solved) problems will be presented at the seminar, starting with the 1-2-3 Conjecture and ending with the Riemann Hypothesis.

1. Jarosław Grytczuk. From the 1-2-3 conjecture to the Riemann hypothesis. European Journal of Combinatorics, 91, 2021, 103213.

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For many hard problems, instead of solving them directly, we need good approximation algorithms. Apart from good their time complexity and decent approximation factor guarantee, we would like to know whether they achieve the best possible approximation ratio (assuming P ≠ NP) possible. Unfortunately, for many NP-complete problems, there is a huge gap between best-known approximation ratio and the ratio that is proved to be unachievable in polynomial time. For instance, for Vertex Cover problem, we don't know any algorithm having a better ratio than 2, and it has been proved in 2005 that it is impossible to get a better ratio than ~1.36. As an attempt to fill in this gap, in 2002, the so-called Unique Games Conjecture was formulated by Khot. It states that having a (1-𝜀)-satisfiable instance of Unique Label Cover problem, it is NP-hard to find a solution satisfying even epsilon fraction of constraints. Assuming it, we are able to prove many tight inapproximability results, for example, it implies that Goemans-Williamson Algorithm for Max-Cut problem achieves the best possible approximation rate. It also follows that we cannot get any better ratio than 2 in the case of Vertex Cover problem. The Unique Games Conjecture is an unusual open problem since the academic world is about evenly divided on whether it is true or not. During the seminar, I will cover this conjecture in more details giving more examples of its influence and presenting recent progress in order to prove it.

1. Subhash Khot. On the Unique Games Conjecture (Invited Survey). 2010 IEEE 25th Annual Conference on Computational Complexity, Cambridge, MA, 2010, pp. 99-121.
2. Bartosz Wodziński. Unique Games Conjecture. slides. 2020.

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Abstract. Suppose that k runners having different constant speeds run laps on a circular track of unit length. The Lonely Runner Conjecture states that, sooner or later, any given runner will be at distance at least 1/k from all the other runners. We prove that with probability tending to one, a much stronger statement holds for random sets in which the bound 1/k is replaced by 1/2 − ε. The proof uses Fourier analytic methods. We also point out some consequences of our result for colouring of random integer distance graphs.

1. Gabriela Czarska. Lonely runner conjecture. slides. 2020.

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Oblivious routing is a routing problem, in which a packet path is selected independently from path choices of other packets. One of the open problems is to find networks for which there exists an oblivious routing algorithm, which allows simultaneously optimizing stretch and congestion of the network. We are presenting an algorithm for oblivious routing on 2dgrid, which is O(log n) approximation for congestion and Θ(1) approximation of stretch.

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In 2013, Dankelmann, Key, and Rodrigues introduced and investigated codes from incidence matrices of a graph. Several authors have been developed their study to several context. In this talk, we present some properties of codes associated with zero divisor graphs. Recall, the zero divisor graph of R denoted by Γ(R), is the simple graph associated with R whose set of vertices consists of all nonzero zero-divisors of R such that two distinct vertices x and y are joined by an edge if xy = 0. This is joint work with K. Abdelmoumen, D. Bennis, and F. Taraza.

1. D. Bennis, J. Mikram, and F. Taraz. On the extended zero divisor graph of commutative rings. Turkish Journal of Mathematics. 40, 376-388, 2016.
2. P. Dankelmann, J. D. Key, and B. G. Rodrigues. Codes from incidence matrices of graphs. Des. Codes Cryptogr. 68, 373-393, 2013.
3. D. F. Anderson, P. S. Livingston. The Zero-Divisor Graph of a Commutative Ring. Journal of Algebra. 217(2), 434-447 , 1999.
4. Raja L'hamri.Codes from zero-divisor graphs. slides. 2020.

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We all know how important mathematical theorems are in general. Less obvious is the fact that theorems in one area like algebra or number theory could have a significant impact on another. In our case, these will be combinatorial problems. In this presentation, We will go through a few simple graph coloring questions (based on the original 1-2-3 Conjecture), which unfortunately don't have simple solutions at all and we'll classify them. Moreover, thanks to Combinatorial Nullstellensatz and some greedy techniques, we will be able to prove some weaker versions of our original claims. And finally, we will see how one simple question, through a chain of small modifications, can lead us to completely different problems.

1. Michał Stobierski. The 1-2-3 Conjecture. slides. 2020.

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The square G² of a graph G is the graph with the same vertex set in which two vertices are joined by an edge if their distance in G is at most two. The chromatic number of the square of a graph G is between D + 1 and D² + 1, where D is the maximum degree of G. Equivalently, the square coloring of a graph is to color the vertices of a graph at distance at most 2 with different colors.

In 1977, Gerd Wegner proved that the square of cubic planar graphs is 8-colorable. He conjectured that his bound can be improved - the chromatic number of G² is at most 7, if D = 3, at most D + 5, if 4 ≤ D ≤ 7, and [3D / 2] + 1, otherwise. Wegner also gave some examples to illustrate that these upper bounds can be obtained.

C. Thomassen (2006) proved the conjecture is true for planar graphs with D = 3. The conjecture is still open for planar graphs with D ≥ 4. However several upper bounds in terms of maximum degree D have been proved as follows. In 1993, Jonas proved that χ(G²) ≤ 9D-19, for planar graphs with D ≥ 5. Agnarsson and Halldorson showed that for every planar graph G with maximum degree D ≥ 749, χ(G²) ≤ [9D / 5] + 2. Van den Heuvel and McGuinness (2003) showed that χ(G²) ≤ 2D + 25, Bordin (2002) proved that χ(G²) ≤ [9D / 5] + 1, if D ≥ 47, and Molloy and Salavatipour (2005) proved χ(G²) ≤ [5D / 3] + 78, moreover, χ(G²) ≤ [5D / 3] + 25 if D ≥ 241. Moreover, conjecture is confirmed in the case of outerplanar graphs and graphs without K₄ minor.

The aim of the seminar will be to present the main facts about Wegner’s conjecture and main techniques and ideas which were used to prove some upper bounds. The presentation will be based on my master thesis.

1. Rafał Byczek. Wegner’s conjecture Colouring the square of a planar graph. slides. 2020.

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Destructive Shift Bribery is a problem in which we are given an election with a set of candidates and a set of voters, a budget , a despised candidate and price for shifting the despised candidate in the voters rankings. Our objective is to ensure that selected candidate cannot win the election. We're going to study the complexity of this problem under diffrent election methods.

1. Andrzej Kaczmarczyk, Piotr Faliszewski. Algorithms for Destructive Shift Bribery. arXiv:1810.01763. 2018.
2. Wojtek Grabis. Conflict-free Replicated Data Types. slides. 2020.

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Sharareh Alipour and Amir Jafari showed various upper bounds for minimal cardinality of (a,b)-dominating set. For positive integers a and b, a subset S ⊆ V(G) is an (a,b)-dominating set if every vertex v ∈ S is adjacent to at least a vertices inside S and every vertex v ∈ V\S is adjacent to at least b vertices inside S. To achieve upper bounds, the authors used Turan's Theorem and Lovasz Local Lemma. These tools allowed them to obtain well-known bounds in a simpler way or new improved bounds in some special cases, including regular graphs.

1. Sharareh Alipour, Amir Jafari. Upper Bounds for the Domination Numbers of Graphs Using Turán’s Theorem and Lovász Local Lemma. Graphs and Combinatorics, 35, 2019, 1153-1160.
2. Jan Mełech. Upper bounds for domination number. slides. 2020.

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3-flow-conjecture
“Every 4-edge-connected graph has a nowhere-zero 3-flow.”

Grötzsch proved that every triangle free (and loopless) planar graph is 3-colorable. By flow/coloring duality, this is equivalent to the statement that every 4-edge-connected planar graph has a nowhere-zero 3-flow. The 3-flow conjecture asserts that this is still true without the assumption of planarity.

Jaeger proved that 4-edge-connected graphs have nowhere-zero 4-flows. The following weak version of the 3-flow conjecture used to remain open until 2010, but the original 3-flow conjecture remains wide open.

C̶o̶n̶j̶e̶c̶t̶u̶r̶e̶ (The weak 3-flow conjecture (Jaeger))
“There exists a fixed integer k so that every k-edge-connected graph has a nowhere-zero 3-flow.” 

These problems and the surrounding results will be presented during the seminar.

1. Carsten Thomassen. The weak 3-flow conjecture and the weak circular flow conjecture. Journal of Combinatorial Theory, Series B, 102(2), 2012, 521-529.
2. Fuyuan Chen, Bo Ning. A note on nowhere-zero 3-flow and Z_3-connectivity. arXiv:1406.1554. 2014.
3. Michał Zwonek. Nowhere Zero Flow and related open problems, slides. 2020.

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If a graph has bounded clique number and sufficiently large chromatic number, what can we say about its induced subgraphs? To answer that question Paul Seymour and Alex Scott took a closer look at recent progress in this field in their χ-boundedness survey. Based on their work I will present some results on forests and holes and few open problems like Gyárfás-Sumner conjecture or χ-boundedness connection to Erdős-Hajnal conjecture.

1. Alex Scott, Paul Seymour. A survey of χ-boundedness. arXiv:1812.07500. 2018.
2. Mateusz Kaczmarek. χ-boundedness, slides. 2020.

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A number is perfect if it is equal to the sum of its divisors. So far only even perfect numbers have been found. For example, it was proven that squares of Mersenne’s numbers are perfect. However, no one has been able to prove that odd perfect numbers don’t exist. I’m going to start by presenting a summary of known facts about odd prime numbers. Then I’ll prove that an odd perfect number with at least eight distinct prime factors has to be divisible by 5.

1. John Voight. On the nonexistence of odd perfect numbers. MASS selecta, 293–300, Amer. Math. Soc., Providence, RI, 2003.
2. Kornel Dulęba. Perfect Numbers. slides. 2020.

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One of number theory open problem is the Lonely Runner Conjecture. It is interesting for several reasons. First the conjecture is relatively intuitive to grasp and easy to state. This conjecture can be find in two different contexts: as a problem in Diophantine’s approximation and as a geometric view obstruction problem. What is more, the difficulty of proving the Lonely Runner Conjecture may seem to increase exponentially with the number of runners. I present statement of the conjecture and known partial results.

1. Clayton Barnes II. The Lonely Runner Conjecture. arXiv:1211.2482. 2012.
2. Thomas W. Cusick, Jorg M. Wills. Lonely runner conjecture. Open Problem Garden. 2007.
3. Bartłomiej Jachowicz. Lonely Runner Conjecture. slides. 2020.

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A cops and robbers problem determines if the number of cops is sufficient to always catch a robber in a game with defined rules played on an undirected graph. Cop number of a graph is the minimal number of cops necessary for cops to win in that game on the specific graph. Mayniel’s conjuncture remains an open problem and states that cop number for graphs of order n is sqrt(n). I’ll present a survey of results achieved that are related to this conjecture.

1. William Baird, Anthony Bonato. Meyniel's conjecture on the cop number: a survey. arXiv:1308.3385. 2013.
2. Filip Bartodziej. Meyniel’s conjecture. slides. 2020.

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I present many results and finally open problem related to synchronizing automata and synchronizing word sends any state of the DFA to one and the same state. This leads to the some natural problems such as: how can we restore control over such a device if we don't know its current state but can observe outputs produced by the device under various actions? I prove some uperbounds for length of this kind of word and in particular I will make a statement of Cerny conjecture.

1. Mikhail V. Volkov. Synchronizing Automata and the Černý Conjecture. LATA 2008. 11-27.
2. Mateusz Pabian. Synchronizing Automata and the Černý Conjecture. slides. 2020.

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The advice model of online computation captures the setting in which the online algorithm is given some partial information concerning the request sequence. We study online computation in a setting in which the advice is provided by an untrusted source. Our objective is to quantify the impact of untrusted advice so as to design and analyze online algorithms that are robust and perform well even when the advice is generated in a malicious, adversarial manner.To this end, we focus on well-studied online problems such as ski rental, online bidding, bin packing, and list update.

1. Spyros Angelopoulos, Christoph Durr, Shendan Jin, Shahin Kamali, and Marc Renault. Online Computation with Untrusted Advice. arXiv:1905.05655. 2019.
2. Adrian Siwiec. Online Computation with Untrusted Advice. slides. 2020.

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Seymour's Second Neighbourhood Conjecture tells us, that any oriented graph has a vertex whose outdegree is at most its second outdegree, which is also known as Second neighborhood problem. Intuitively, it suggests that in a social network described by such a graph, someone will have at least as many friends-of-friends as friends. We will say about Chen-Shen-Yuster prove, that for any digraph D, there exists a vertex v such that |N⁺⁺(v)|≥γ|N⁺(v)|, where γ=0.67815. We will consider graphs, in which we know, that such vertex exists. We will also say about unsuccessful attempts at proving this conjecture.

1. Salman Ghazal. Seymour's second neighborhood conjecture for tournaments missing a generalized star. arXiv:1106.0085, 2011.
2. Tyler Seacrest. Seymour's Second Neighborhood Conjecture for Subsets of Vertices. arXiv:1106.0085, 2018.
3. Wojciech Buczek. Seymour’s Second Neighbourhood Conjecture. slides. 2020.

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The Collatz conjecture, also known as 3n + 1 conjecture considers a function, which returns n/2 if n is even and 3n + 1 if n is odd. The conjecture states that for every n we can repeatedly apply this function to eventually reach 1. I will talk about different approaches to proving this conjecture.

1. Mikołaj Twaróg. Collatz conjecture. slides. 2020.

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The game of edge geography is played by two players who alternately move a token on a graph from one vertex to an adjacent vertex, erasing the edge in between. The player who first has no legal move loses the game. We analyze the space complexity of the decision problem of determining whether a start position in this game is a win for the first player. We also show a polynomial time algorithm for finding winning moves for bipartite graphs.

1. Aviezri S. Fraenkel, Edward R. Scheinerman, Daniel Ullman. Undirected Edge Geography. Theoretical Computer Science, 112(2), 1993, 371-381.
2. Adam Pardyl. Undirected edge geography. slides. 2020.

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Consider a directed graph such that every vertex has at most 2 outgoing edges - one of them is labeled as 'open' (we can traverse it) and the second one is labeled as 'closed' (we cannot traverse it). Every time we go somewhere from the vertex v, labels at its two edges are swapped, so the next time we visit v, we will take another direction. Given designated two vertices: origin and destination, we need to decide, whether eventually we will reach destination starting in the origin. This problem lies in both NP and coNP, but it is still an open question whether it belongs to P.

1. Jérôme Dohrau, Bernd Gärtner, Manuel Kohler, Jiří Matoušek, Emo Welzl. ARRIVAL: A zero-player graph game in NP ∩ coNP. arXiv:1605.03546. 2020.
2. Piotr Mikołajczyk. ARRIVAL game. slides. 2020.

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Let P be a set of n points in R². A point q (not necessarily in P) is called a centerpoint of P if each closed half-plane containing q at least ⌈n/3⌉ points of P, or, equivalently, any convex set that contains more than ²/₃ n points of P must also contain q. It is a well-known fact that a centerpoint always exists and the constant ²/₃ is the best possible. Can we improve this constant by using, say, two points, or some other small number of points? In the presentation we'll try to answer those questions.

Vladyslav Rachek. Small weak epsilon-nets. slides. 2020.

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The aim of this pearl is to solve the following well-known problem that appears in many puzzle books, for example Levitin & Levitin (2011) and Bellos (2016), usually for the particular case n=12.

There are n coins, all identical in appearance, one of which may be fake. The fake coin, if it exists, is either lighter or heavier than the fair coins, but it is not known which, nor by how much. Given is a balance with two pans but no weights. Equal number of coins can be placed on each pan and weighed. There are three possible outcomes: the left-hand pan may be lighter than the right-hand pan, or of equal weight, or heavier. Design a recipe for determining the minimum number of weighings guaranteed to determine the fake coin, if it exists, and whether it is lighter or heavier than the others.

During today presentation we will learn how we can use graph theory to proof hardness of general problem of manipulating poll outcome. Based on paper: "Selecting Voting Locations for Fun and Profit" written by Zack Fitzsimmons and Omer Lev.

Zack Fitzsimmons, Omer Lev. Selecting Voting Locations for Fun and Profit. arXiv:2003.06879. 2020.

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We will focus on Hadwiger-Nelson problem - an open question from geometric graph theory that asks for the minimum number of colors required to color the plane such that no two points at distance 1 from each other have the same color. There are a few interesting theorems related to the problem and results which we will go through. We will focus in particular on the most recent result of Aubrey de Grey who showed that the desired chromatic number is at least 5.

1. Aubrey D.N.J. de Grey. The chromatic number of the plane is at least 5. arXiv:1804.02385. 2018.
2. Mateusz Tokarz. slides. 2020.

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In the paper authors consider certain variants of Graph Isomorphism problem. They focus on properties of graph spectra and eigenspaces - namely if Laplacian of one of the graphs is greater or equal to another in Loewner ordering. In the first part of the paper they prove that one of the problems named Spectral Graph Dominance is NPC. The rest of the paper is devoted to an approximation algorithm for special case of the problem called Spectrally Robust Graph Isomorphism.

Alexandra Kolla, Ioannis Koutis, Vivek Madan, Ali Kemal Sinop. Spectrally Robust Graph Isomorphism. arXiv:1805.00181, 1-17, 2018.

Authors study Uber's pricing mechanisms from the perspective of drivers, presenting the theoretical foundation that has informed the design of Uber’s new additive driver surge mechanism. They present a dynamic stochastic model to capture the impact of surge pricing on driver earnings and their strategies to maximize such earnings.

Nikhil Garg, Hamid Nazerzadeh. Driver Surge Pricing. arXiv. 2019.

The paper presents proof that the queue number of planar graphs is bounded. It also mentions generalizations of the result and other theorems that have similar proofs.

Vida Dujmović, Gwenaël Joret, Piotr Micek, Pat Morin, Torsten Ueckerdt, David R. Wood. Planar graphs have bounded queue-number. arXiv. 2019.

The notion of nowhere dense classes of graphs attracted much attention in recent years and found many applications in structural graph theory and algorithmics. The powers of nowhere dense graphs do not need to be sparse, for instance the second power of star graphs are complete graphs. However, it is believed that powers of sparse graphs inherit somewhat simple structure. In this spirit, we show that for a fixed nowhere dense class of graphs, a positive real ε and a positive integer d, in any n-vertex graph G in the class, there are disjoint vertex subsets A and B with |A|=Ω(n) and |B|=Ω(n^1-ε) such that in the d-th power of G, either there is no edge between A and B, or there are all possible edges between A and B.

Destructive Shift Bribery is a problem in which we are given an election with a set of candidates and a set of voters, a budget , a despised candidate and price for shifting the despised candidate in the voters rankings. Our objective is to ensure that selected candidate cannot win the election. We're going to study the complexity of this problem under diffrent election methods.

Andrzej Kaczmarczyk, Piotr Faliszewski. Algorithms for Destructive Shift Bribery. arXiv. 2018.

The paper shows that the Diffie-Hellman protocol is not as secure as we thought. The authors present the Logjam attack, which consists in quickly calculating discrete logarithms based on previously performed calculations. This can be done because many websites use the same prime numbers in the message encryption process.

David Adrian, Karthikeyan Bhargavan, Zakir Durumeric, Pierrick Gaudry, Matthew Green, J. Alex Halderman, Nadia Heninger, Drew Springall, Emmanuel Thomé, Luke Valenta, Benjamin VanderSloot, Eric Wustrow, Santiago Zanella-Béguelink, Paul Zimmermann. Imperfect Forward Secrecy: How Diffie-Hellman Fails in Practice. WeakDH.org.

he consensus problem involves an asynchronous system of processes, some of which may be unreliable. The problem is for reliable processes to agree on a binary value. In this paper, it is shown that every protocol for this problem has the possibility of nontermination, even with only one faulty process. By way of contrast, solutions are known for the synchronous case, the “Byzantine Generals” problem.

Authors of the paper were awarded a Dijkstra Prize for this work - given to the most influential papers in distributed computing.

Michael J. Fischer, Nancy A. Lynch, Michael S. Paterson. Impossibility of Distributed Consensus with One Faulty Process. Journal of the Association for Computing Machinery, 32(2), 1985, 374-382.

Unevenly cut pizza is a frustrating occurrence. How can we then make sure that a friend is not trying to reduce our portion of the delicious meal? We will present a strategy which guarantees that one will leave the table satisfied, assuming that they started eating first.

Kolja Knauer, Piotr Micek, Torsten Ueckerdt. How to eat 4/9 of a pizza. arXiv. 2008.

I will present a proof that there exists a function f(d), such that there exists a 3-coloring of any planar graph G in which each monochromatic subgraph has at most f(d) vertices, where d is the degree of the highest-degree vertex in G.

Louis Esperet, Gwenaël Joret. Coloring planar graphs with three colors and no large monochromatic components. arXiv, 1303.2487. 2013.

Survey of Hadwiger's Conjecture from 1943, that for all t ≥ 0, every graph is either t-colorable or has a subgraph that can be contracted to the complete t+1 vertices graph. This conjecture is the tremendous strengthening of the four-color problem also known as map coloring problem.

Paul Seymour. Hadwiger’s conjecture.

The most studied linear algebraic operation, matrix multiplication, has surprisingly fast O(n^ω) time algorithms, for ω<2.373. On the other hand, the (min,+)-product, which is at the heart of APSP, is widely conjectured to require cubic time. There is a plethora of matrix products and graph problems whose complexity seems to lie in the middle of these two problems, e.g. the (min,max)-product, the min-witness product, APSP in directed unweighted graphs. The best runtimes for these "intermediate" problems are all O(n^(3+ω)/2). A similar phenomenon occurs for convolution problems.