Discrete Mathematics Group

Chong Shangguan (上官冲), The hat guessing number of graphs
2023-06-27 16:30 - 17:30 (Asia/Seoul) | Room B332, IBS (기초과학연구원)
Consider the following hat guessing game: $n$ players are placed on $n$ vertices of a graph, each wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. Given a graph $G$, its hat guessing number $HG(G)$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
In 2008, Butler, Hajiaghayi, Kleinberg, and Leighton asked whether the hat guessing number of the complete bipartite graph $K_{n,n}$ is at least some fixed positive (fractional) power of $n$. We answer this question affirmatively, showing that for sufficiently large $n$, $HG(K_{n,n})ge n^{0.5-o(1)}$. Our guessing strategy is based on some ideas from coding theory and probabilistic method.
Based on a joint work with Noga Alon, Omri Ben-Eliezer, and Itzhak Tamo.

Guanghui Wang (王光辉), Embeddings in uniformly dense hypergraphs
2023-06-20 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
An archetype problem in extremal combinatorics is to study the structure of subgraphs appearing in different classes of (hyper)graphs. We will focus on such embedding problems in uniformly dense hypergraphs. In precise, we will mention the uniform Turan density of some hypergraphs.

Minho Cho (조민호), Strong Erdős-Hajnal property on chordal graphs and its variants
2023-06-13 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
A graph class $mathcal{G}$ has the strong Erdős-Hajnal property (SEH-property) if there is a constant $c=c(mathcal{G}) > 0$ such that for every member $G$ of $mathcal{G}$, either $G$ or its complement has $K_{m, m}$ as a subgraph where $m geq leftlfloor c|V(G)| rightrfloor$. We prove that the class of chordal graphs satisfies SEH-property with constant $c = 2/9$.
On the other hand, a strengthening of SEH-property which we call the colorful Erdős-Hajnal property was discussed in geometric settings by Alon et al.(2005) and by Fox et al.(2012). Inspired by their results, we show that for every pair $F_1, F_2$ of subtree families of the same size in a tree $T$ with $k$ leaves, there exist subfamilies $F’_1 subseteq F_1$ and $F’_2 subseteq F_2$ of size $theta left( frac{ln k}{k} left| F_1 right|right)$ such that either every pair of representatives from distinct subfamilies intersect or every such pair do not intersect. Our results are asymptotically optimal.
Joint work with Andreas Holmsen, Jinha Kim and Minki Kim.

Suyun Jiang (江素云), How connectivity affects the extremal number of trees
2023-05-30 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
The Erdős-Sós conjecture states that the maximum number of edges in an $n$-vertex graph without a given $k$-vertex tree is at most $frac {n(k-2)}{2}$. Despite significant interest, the conjecture remains unsolved. Recently, Caro, Patkós, and Tuza considered this problem for host graphs that are connected. Settling a problem posed by them, for a $k$-vertex tree $T$, we construct $n$-vertex connected graphs that are $T$-free with at least $(1/4-o_k(1))nk$ edges, showing that the additional connectivity condition can reduce the maximum size by at most a factor of 2. Furthermore, we show that this is optimal: there is a family of $k$-vertex brooms $T$ such that the maximum size of an $n$-vertex connected $T$-free graph is at most $(1/4+o_k(1))nk$.

Szymon Toruńczyk, Flip-width: Cops and Robber on dense graphs
2023-05-17 16:00 - 17:00 | Zoom ID: 869 4632 6610 (ibsdimag)
We define new graph parameters, called flip-width, that generalize treewidth, degeneracy, and generalized coloring numbers for sparse graphs, and clique-width and twin-width for dense graphs. The flip-width parameters are defined using variants of the Cops and Robber game, in which the robber has speed bounded by a fixed constant r∈N∪{∞}, and the cops perform flips (or perturbations) of the considered graph. We then propose a new notion of tameness of a graph class, called bounded flip-width, which is a dense counterpart of classes of bounded expansion of Nešetril and Ossona de Mendez, and includes classes of bounded twin-width of Bonnet, Kim, Thomassé, and Watrigant. This unifies Sparsity Theory and Twin-width Theory, providing a common language for studying the central notions of the two theories, such as weak coloring numbers and twin-width — corresponding to winning strategies of one player — or dense shallow minors, rich divisions, or well-linked sets, corresponding to winning strategies of the other player. We prove that boundedness of flip-width is preserved by first-order interpretations, or transductions, generalizing previous results concerning classes of bounded expansion and bounded twin-width. We provide an algorithm approximating the flip-width of a given graph, which runs in slicewise polynomial time (XP) in the size of the graph. Finally, we propose a more general notion of tameness, called almost bounded flip-width, which is a dense counterpart of nowhere dense classes. We conjecture, and provide evidence, that classes with almost bounded flip-width coincide with monadically dependent (or monadically NIP) classes, introduced by Shelah in model theory. We also provide evidence that classes of almost bounded flip-width characterise the hereditary graph classes for which the model-checking problem is fixed-parameter tractable.

Oliver Janzer, Small subgraphs with large average degree
2023-05-16 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
We study the fundamental problem of finding small dense subgraphs in a given graph. For a real number $s>2$, we prove that every graph on $n$ vertices with average degree at least $d$ contains a subgraph of average degree at least $s$ on at most $nd^{-frac{s}{s-2}}(log d)^{O_s(1)}$ vertices. This is optimal up to the polylogarithmic factor, and resolves a conjecture of Feige and Wagner. In addition, we show that every graph with $n$ vertices and average degree at least $n^{1-frac{2}{s}+varepsilon}$ contains a subgraph of average degree at least $s$ on $O_{varepsilon,s}(1)$ vertices, which is also optimal up to the constant hidden in the $O(.)$ notation, and resolves a conjecture of Verstraëte.
Joint work with Benny Sudakov and Istvan Tomon.

Maria Chudnovsky, Induced subgraphs and tree decompositions
2023-05-11 16:15 - 17:15 | Room 1501, Bldg. E6-1, KAIST
Tree decompositions are a powerful tool in both structural graph theory and graph algorithms. Many hard problems become tractable if the input graph is known to have a tree decomposition of bounded “width”. Exhibiting a particular kind of a tree decomposition is also a useful way to describe the structure of a graph. Tree decompositions have traditionally been used in the context of forbidden graph minors; bringing them into the realm of forbidden induced subgraphs has until recently remained out of reach. Over the last couple of years we have made significant progress in this direction, exploring both the classical notion of bounded tree-width, and concepts of more structural flavor. This talk will survey some of these ideas and results.

Jozef Skokan, Separating the edges of a graph by a linear number of paths
2023-05-09 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Recently, Letzter proved that any graph of order n contains a collection P of $O(n log^*n)$ paths with the following property: for all distinct edges e and f there exists a path in P which contains e but not f. We improve this upper bound to 19n, thus answering a question of Katona and confirming a conjecture independently posed by Balogh, Csaba, Martin, and Pluhar and by Falgas-Ravry, Kittipassorn, Korandi, Letzter, and Narayanan.
Our proof is elementary and self-contained.

Rob Morris, An exponential improvement for diagonal Ramsey
2023-05-02 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
The Ramsey number $R(k)$ is the minimum n such that every red-blue colouring of the edges of the complete graph on n vertices contains a monochromatic copy of $K_k$. It has been known since the work of Erdős and Szekeres in 1935, and Erdős in 1947, that $2^{k/2} < R(k) < 4^k$, but in the decades since the only improvements have been by lower order terms. In this talk I will sketch the proof of a very recent result, which improves the upper bound of Erdős and Szekeres by a (small) exponential factor.
Based on joint work with Marcelo Campos, Simon Griffiths and Julian Sahasrabudhe.

Rob Morris, Ramsey theory: searching for order in chaos
2023-04-27 16:15 - 17:15 | Room 1501, Bldg. E6-1, KAIST
In many different areas of mathematics (such as number theory, discrete geometry, and combinatorics), one is often presented with a large “unstructured” object, and asked to find a smaller “structured” object inside it. One of the earliest and most influential examples of this phenomenon was the theorem of Ramsey, proved in 1930, which states that if n = n(k) is large enough, then in any red-blue colouring of the edges of the complete graph on n vertices, there exists a monochromatic clique on k vertices. In this talk I will discuss some of the questions, ideas, and new techniques that were inspired by this theorem, and present some recent progress on one of the central problems in the area: bounding the so-called “diagonal” Ramsey numbers. Based on joint work with Marcelo Campos, Simon Griffiths and Julian Sahasrabudhe.

Hyunwoo Lee (이현우), On perfect subdivision tilings
2023-04-25 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
For a given graph $H$, we say that a graph $G$ has a perfect $H$-subdivision tiling if $G$ contains a collection of vertex-disjoint subdivisions of $H$ covering all vertices of $G.$ Let $delta_{sub}(n, H)$ be the smallest integer $k$ such that any $n$-vertex graph $G$ with minimum degree at least $k$ has a perfect $H$-subdivision tiling. For every graph $H$, we asymptotically determined the value of $delta_{sub}(n, H)$. More precisely, for every graph $H$ with at least one edge, there is a constant $1 < xi^*(H)leq 2$ such that $delta_{sub}(n, H) = left(1 - frac{1}{xi^*(H)} + o(1) right)n$ if $H$ has a bipartite subdivision with two parts having different parities. Otherwise, the threshold depends on the parity of $n$.

Shin-ichiro Seki, On the extension of the Green-Tao theorem to number fields
2023-04-19 16:30 - 17:30 | Zoom ID: 897 6822 0619 (ibsecopro) [04/19 only]
In 2006, Tao established the Gaussian counterpart of the celebrated Green-Tao theorem on arithmetic progressions of primes. In this talk, I will explain the extension of Tao’s theorem and the Green-Tao theorem to the case of general number fields. Our combinatorial tool is the relative hypergraph removal lemma by Conlon-Fox-Zhao. I will discuss the difficulties that arise in the case of general number fields and an application of our results to prime representations by binary quadratic forms. This is based on joint work with Wataru Kai, Masato Mimura, Akihiro Munemasa, and Kiyoto Yoshino.

MATRIX-IBS Workshop: Structural Graph Theory Downunder III
2023-04-17 - 2023-04-24 | MATRIX, Australia
Website: https://www.matrix-inst.org.au/events/structural-graph-theory-downunder-iii/
Program Description:
This program, jointly organised by MATRIX and the Discrete Mathematics Group of the Korean Institute for Basic Science (IBS), builds on the “Structural Graph Theory Downunder” programs held at MATRIX in 2019 and 2022. In this short intensive workshop, mathematicians from across the globe will come together to work on open problems in structural graph theory. The focus will be on graph colouring, graph minors, induced subgraphs, and graph product structure theory. The majority of the time will be allocated to collaborative research, with only a few talks describing recent advances. The goal is to create an environment where mathematicians at all career stages work side-by-side. We anticipate that open problems will be solved, and lasting collaborations will be initiated.
Confirmed Participants:
Rutger Campbell (Institute for Basic Science, Korea)
Linda Cook (Institute for Basic Science, Korea)
James Davies (University of Cambridge, UK)
Marc Distel (Monash University, Australia)
Zdeněk Dvořák (Charles University, Czech Republic)
Bryce Frederickson (Emory University, USA)
António Girão (University of Oxford, UK)
Pascal Gollin (Institute for Basic Science, Korea)
Kevin Hendrey (Institute for Basic Science, Korea)
Robert Hickingbotham (Monash University, Australia)
Freddie Illingworth (University of Oxford, UK)
O-joung Kwon (Hanyang University, Korea)
Florian Lehner (University of Auckland, New Zealand)
Anita Liebenau (UNSW, Australia)
Chun-Hung Liu (Texas A&M, USA)
Rose McCarty (Princeton University, USA)
Lukas Michel (University of Oxford, UK)
Sang-il Oum (Institute for Basic Science and KAIST, Korea)
Michael Savery (University of Oxford, UK)
Alex Scott (University of Oxford, UK)
Raphael Steiner (ETH Zurich, Switzerland)
Jane Tan (University of Oxford, UK)
Sebastian Wiederrecht (Institute for Basic Science, Korea)
David Wood (Monash University, Australia)
Liana Yepremyan (Emory University, USA)

James Davies, Two structural results for pivot-minors
2023-04-11 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Pivot-minors can be thought of as a dense analogue of graph minors. We shall discuss pivot-minors and two recent results for proper pivot-minor-closed classes of graphs. In particular, that for every graph H, the class of graphs containing no H-pivot-minor is 𝜒-bounded, and also satisfies the (strong) Erdős-Hajnal property.

Jie Han, Spanning trees in expanders
2023-04-06 10:00 - 11:00 | Zoom ID: 224 221 2686 (ibsecopro)
We consider the spanning tree embedding problem in dense graphs without bipartite holes and sparse graphs. In 2005, Alon, Krivelevich and Sudakov asked for determining the best possible spectral gap forcing an $(n,d,lambda)$-graph to be $T(n, Delta)$-universal. In this talk, we introduce our recent work on this conjecture.

István Tomon, Configurations of boxes
2023-04-04 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Configurations of axis-parallel boxes in $mathbb{R}^d$ are extensively studied in combinatorial geometry. Despite their perceived simplicity, there are many problems involving their structure that are not well understood. I will talk about a construction that shows that their structure might be more complicated than people conjectured.

Tianchi Yang, On the maximum number of edges in k-critical graphs
2023-03-28 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
In this talk, we will discuss the problem of determining the maximum number of edges in an n-vertex k-critical graph. A graph is considered k-critical if its chromatic number is k, but any proper subgraph has a chromatic number less than k. The problem remains open for any integer k ≥ 4. Our presentation will showcase an improvement on previous results achieved by employing a combination of extremal graph theory and structural analysis. We will introduce a key lemma, which may be of independent interest, as it sheds light on the partial structure of dense k-critical graphs. This is joint work with Cong Luo and Jie Ma.

Qizhong Lin, Two classical Ramsey-Turán numbers involving triangles
2023-03-22 16:30 - 17:30 | Zoom ID: 224 221 2686 (ibsecopro)
In 1993, Erdős, Hajnal, Simonovits, Sós and Szemerédi proposed to determine the value of Ramsey-Turán density $rho(3,q)$ for $qge3$. Erdős et al. (1993) and Kim, Kim and Liu (2019) proposed two corresponding conjectures. However, we only know four values of this Ramsey-Turán density by Erdős et al. (1993). There is no progress on this classical Ramsey-Turán density since then. In this talk, I will introduce two new values of this classical Ramsey-Turán density. Moreover, the corresponding asymptotically extremal structures are weakly stable, which answers a problem of Erdős et al. (1993) for the two cases. Joint work with Xinyu Hu.

Younjin Kim (김연진), Problems on Extremal Combinatorics
2023-03-21 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Extremal Combinatorics studies the maximum or minimum size of finite objects (numbers, sets, graphs) satisfying certain properties. In this talk, I introduce the conjectures I solved on Extremal Combinatorics, and also introduce recent extremal problems.

Paul Seymour, A loglog step towards the Erdős-Hajnal conjecture
2023-03-16 10:00 - 11:00 | Zoom ID: 869 4632 6610 (ibsdimag)
In 1977, Erdős and Hajnal made the conjecture that, for every graph $H$, there exists $c>0$ such that every $H$-free graph $G$ has a clique or stable set of size at least $|G|^c$; and they proved that this is true with $|G|^c$ replaced by $2^{csqrt{log |G|}}$.
There has no improvement on this result (for general $H$) until now, but now we have an improvement: that for every graph $H$, there exists $c>0$ such that every $H$-free graph $G$ with $|G|ge 2$ has a clique or stable set of size at least $$2^{csqrt{log |G|loglog|G|}}.$$ This talk will outline the proof.
Joint work with Matija Bucić, Tung Nguyen and Alex Scott.

Stijn Cambie, Recent progress on the Union-closed conjecture and related
2023-03-14 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
We give a summary on the work of the last months related to Frankl’s Union-Closed conjecture and its offsprings. The initial conjecture is stated as a theorem in extremal set theory; when a family F is union-closed (the union of sets of F is itself a set of $mathcal F$), then $mathcal F$ contains an (abundant) element that belongs to at least half of the sets. Meanwhile, there are many related versions of the conjecture due to Frankl. For example, graph theorists may prefer the equivalent statement that every bipartite graph has a vertex that belongs to no more than half of the maximal independent sets. While even proving the ε-Union-Closed Sets Conjecture was out of reach, Poonen and Cui & Hu conjectured already stronger forms.
At the end of last year, progress was made on many of these conjectures. Gilmer proved the ε-Union-Closed Sets Conjecture using an elegant entropy-based method which was sharpened by many others. Despite a sharp approximate form of the union-closed conjecture as stated by Chase and Lovett, a further improvement was possible. In a different direction, Kabela, Polak and Teska constructed union-closed family sets with large sets and few abundant elements.
This talk will keep the audience up-to-date and give them insight in the main ideas.
People who would like more details, can join the Ecopro-reading session on the 14th of March (10 o’clock, B332) as well. Here we go deeper in the core of the proofs and discuss possible directions for the full resolution.

Marcelo Sales, On Pisier type problems
2023-03-09 10:00 - 11:00 | Zoom ID: 224 221 2686 (ibsecopro)
A subset $Asubseteq mathbb Z$ of integers is free if for every two distinct subsets $B, B’subseteq A$ we have [ sum_{bin B}b
eq sum_{b’in B’} b’.]Pisier asked if for every subset $Asubseteq mathbb Z$ of integers the following two statement are equivalent:
(i) $A$ is a union of finitely many free sets.
(ii) There exists $epsilon>0$ such that every finite subset $Bsubseteq A$ contains a free subset $Csubseteq B$ with $|C|geq epsilon |B|$.
In a more general framework, the Pisier question can be seen as the problem of determining if statements (i) and (ii) are equivalent for subsets of a given structure with prescribed property. We study the problem for several structures including $B_h$-sets, arithmetic progressions, independent sets in hypergraphs and configurations in the euclidean space. This is joint work with Jaroslav Nešetřil and Vojtech Rödl.

Eunjin Oh (오은진), Parameterized algorithms for the planar disjoint paths problem
2023-03-07 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Given an undirected planar graph $G$ with $n$ vertices and a set $T$ of $k$ pairs $(s_i,t_i)_{i=1}^k$ of vertices, the goal of the planar disjoint paths problem is to find a set $mathcal P$ of $k$ pairwise vertex-disjoint paths connecting $s_i$ and $t_i$ for all indices $iin{1,ldots,k}$. This problem has been studied extensively due to its numerous applications such as VLSI layout and circuit routing. However, this problem is NP-complete even for grid graphs. This motivates the study of this problem from the viewpoint of parameterized algorithms.
In this talk, I will present a $2^{O(k^2)}n$-time algorithm for the planar disjoint paths problem. This improves the two previously best-known algorithms: $2^{2^{O(k)}}n$-time algorithm [Discrete Applied Mathematics 1995] and $2^{O(k^2)}n^6$-time algorithm [STOC 2020].
This is joint work with Kyungjin Cho and Seunghyeok Oh.

Maya Sankar, The Turán Numbers of Homeomorphs
2023-02-28 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Let $X$ be a 2-dimensional simplicial complex. Denote by $text{ex}_{hom}(n,X)$ the maximum number of 2-simplices in an $n$-vertex simplicial complex that has no sub-simplicial complex homeomorphic to $X$. The asymptotics of $text{ex}_{hom}(n,X)$ have recently been determined for all surfaces $X$. I will discuss these results, as well as ongoing work bounding $text{ex}_{hom}(n,X)$ for arbitrary 2-dimensional simplicial complexes $X$.

Daniel Altman, On an arithmetic Sidorenko conjecture, and a question of Alon
2023-02-22 17:00 - 18:00 | Zoom ID: 224 221 2686 (ibsecopro)
Let $G=mathbb{F}_p^n$. Which systems of linear equations $Psi$ have the property that amongst all subsets of $G$ of fixed density, random subsets minimise the number of solutions to $Psi$? This is an arithmetic analogue of a well-known conjecture of Sidorenko in graph theory, which has remained open and of great interest since the 1980s. We will discuss some recent results along these lines, with particular focus on some of the ideas behind a negative answer to a related question of Alon.

Meike Hatzel, Fixed-Parameter Tractability of Directed Multicut with Three Terminal Pairs Parametrised by the Size of the Cutset: Twin-Width Meets Flow-Augmentation
2023-02-21 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
We show fixed-parameter tractability of the Directed Multicut problem with three terminal pairs (with a randomized algorithm). This problem, given a directed graph $G$, pairs of vertices (called terminals) $(s_1,t_1)$, $(s_2,t_2)$, and $(s_3,t_3)$, and an integer $k$, asks to find a set of at most $k$ non-terminal vertices in $G$ that intersect all $s_1t_1$-paths, all $s_2t_2$-paths, and all $s_3t_3$-paths. The parameterized complexity of this case has been open since Chitnis, Cygan, Hajiaghayi, and Marx proved fixed-parameter tractability of the 2-terminal-pairs case at SODA 2012, and Pilipczuk and Wahlström proved the W[1]-hardness of the 4-terminal-pairs case at SODA 2016.
On the technical side, we use two recent developments in parameterized algorithms. Using the technique of directed flow-augmentation [Kim, Kratsch, Pilipczuk, Wahlström, STOC 2022] we cast the problem as a CSP problem with few variables and constraints over a large ordered domain. We observe that this problem can be in turn encoded as an FO model-checking task over a structure consisting of a few 0-1 matrices. We look at this problem through the lenses of twin-width, a recently introduced structural parameter [Bonnet, Kim, Thomassé, Watrigant, FOCS 2020]: By a recent characterization [Bonnet, Giocanti, Ossona de Mendez, Simon, Thomassé, Toruńczyk, STOC 2022] the said FO model-checking task can be done in FPT time if the said matrices have bounded grid rank. To complete the proof, we show an irrelevant vertex rule: If any of the matrices in the said encoding has a large grid minor, a vertex corresponding to the “middle” box in the grid minor can be proclaimed irrelevant — not contained in the sought solution — and thus reduced.

Robert Hickingbotham, Treewidth, Circle Graphs and Circular Drawings
2023-02-15 16:30 - 17:30 | Zoom ID: 869 4632 6610 (ibsdimag)
A circle graph is an intersection graph of a set of chords of a circle. In this talk, I will describe the unavoidable induced subgraphs of circle graphs with large treewidth. This includes examples that are far from the `usual suspects’. Our results imply that treewidth and Hadwiger number are linearly tied on the class of circle graphs, and that the unavoidable induced subgraphs of a vertex-minor-closed class with large treewidth are the usual suspects if and only if the class has bounded rank-width. I will also discuss applications of our results to the treewidth of graphs $G$ that have a circular drawing whose crossing graph is well-behaved in some way. In this setting, our results show that if the crossing graph is $K_t$-minor-free, then $G$ has treewidth at most $12t-23$ and has no $K_{2,4t}$-topological minor. On the other hand, I’ll present a construction of graphs with arbitrarily large Hadwiger number that have circular drawings whose crossing graphs are $2$-degenerate. This is joint work with Freddie Illingworth, Bojan Mohar, and David R. Wood

Raphael Steiner, Strengthening Hadwiger's conjecture for 4- and 5-chromatic graphs
2023-02-14 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Hadwiger’s famous coloring conjecture states that every t-chromatic graph contains a $K_t$-minor. Holroyd [Bull. London Math. Soc. 29, (1997), pp. 139-144] conjectured the following strengthening of Hadwiger’s conjecture: If G is a t-chromatic graph and S⊆V(G) takes all colors in every t-coloring of G, then G contains a $K_t$-minor rooted at S. We prove this conjecture in the first open case of t=4. Notably, our result also directly implies a stronger version of Hadwiger’s conjecture for 5-chromatic graphs as follows: Every 5-chromatic graph contains a $K_5$-minor with a singleton branch-set. In fact, in a 5-vertex-critical graph we may specify the singleton branch-set to be any vertex of the graph. Joint work with Anders Martinsson (ETH).

Benjamin Bergougnoux, Tight Lower Bounds for Problems Parameterized by Rank-width
2023-02-01 16:30 - 17:30 | Zoom ID: 869 4632 6610 (ibsdimag)
We show that there is no $2^{o(k^2)} n^{O(1)}$ time algorithm for Independent Set on $n$-vertex graphs with rank-width $k$, unless the Exponential Time Hypothesis (ETH) fails. Our lower bound matches the $2^{O(k^2)} n^{O(1)}$ time algorithm given by Bui-Xuan, Telle, and Vatshelle [Discret. Appl. Math., 2010] and it answers the open question of Bergougnoux and Kanté [SIAM J. Discret. Math., 2021]. We also show that the known $2^{O(k^2)} n^{O(1)}$ time algorithms for Weighted Dominating Set, Maximum Induced Matching and Feedback Vertex Set parameterized by rank-width $k$ are optimal assuming ETH. Our results are the first tight ETH lower bounds parameterized by rank-width that do not follow directly from lower bounds for $n$-vertex graphs.
This is a joint work with Tuukka Korhonen and Jesper Nederlof.
Accepted to STACS 2023 and available on arXiv https://arxiv.org/abs/2210.02117

Abhishek Methuku, A proof of the Erdős–Faber–Lovász conjecture
2023-01-31 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
The Erdős–Faber–Lovász conjecture (posed in 1972) states that the chromatic index of any linear hypergraph on n vertices is at most n. In this talk, I will sketch a proof of this conjecture for every large n. Joint work with D. Kang, T. Kelly, D. Kühn and D. Osthus.
