Discrete Mathematics Group

Ron Aharoni, A strong version of the Caccetta-Haggkvist conjecture
2022-01-13 16:30 - 17:30 (Asia/Seoul) | Zoom ID: 869 4632 6610 (ibsdimag)
The Caccetta-Haggkvist conjecture, one of the best known in graph theory, is that in a digraph with $n$ vertices in which all outdegrees are at least $n/k$ there is a directed cycle of length at most $k$. This is known for large values of $k$, relatively to n, and asymptotically for n large. A few years ago I offered a generalization: given sets $F_1$, $ldots$, $F_n$ of sets of undirected edges, each of size at least $n/k$, there exists a rainbow undirected cycle of length at most $k$. The directed version is obtained by taking as $F_i$ the set of edges going out of the vertex $v_i$ ($i le n$), with the directions removed. I will tell about recent results on this conjecture, obtained together with He Guo, with Beger, Chudnovsky and Zerbib, and with DeVos and Holzman.

Andreas Holmsen, Some recent results on geometric transversals
2022-01-11 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
A geometric transversal to a family of convex sets in $mathbb R^d$ is an affine flat that intersects the members of the family. While there exists a far-reaching theory concerning 0-dimensional transversals (intersection patterns of convex sets), much less is known when it comes to higher-dimensional transversals. In this talk, I will present some new and old results and problems regarding geometric transversals, based on joint work with Otfried Cheong and Xavier Goaoc.

Seunghun Lee (이승훈), Transversals and colorings of simplicial spheres
2022-01-04 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
Motivated from the surrounding property of a point set in $mathbb{R}^d$ introduced by Holmsen, Pach and Tverberg, we consider the transversal number and chromatic number of a simplicial sphere. As an attempt to give a lower bound for the maximum transversal ratio of simplicial $d$-spheres, we provide two infinite constructions. The first construction gives infinitely many $(d+1)$-dimensional simplicial polytopes with the transversal ratio exactly $frac{2}{d+2}$ for every $dgeq 2$. In the case of $d=2$, this meets the previously well-known upper bound $1/2$ tightly. The second gives infinitely many simplicial 3-spheres with the transversal ratio greater than $1/2$. This was unexpected from what was previously known about the surrounding property. Moreover, we show that, for $dgeq 3$, the facet hypergraph $mathcal{F}(mathbf{P})$ of a $(d+1)$-dimensional simplicial polytope $mathbf{P}$ has the chromatic number $chi(mathcal{F}(mathbf{P})) in O(n^{frac{lceil d/2rceil-1}{d}})$, where $n$ is the number of vertices of $mathbf{P}$. This slightly improves the upper bound previously obtained by Heise, Panagiotou, Pikhurko, and Taraz. This is a joint work with Joseph Briggs and Michael Gene Dobbins.

2021 Combinatorics Workshop (2021 조합론 학술대회)
2021-12-20 - 2021-12-22 | The Bloomvista
The Combinatorics Workshop (조합론 학술대회) is an annual conference of combinatorialists in Korea that started in 2004 by the Yonsei University BK21 Research Group. Since 2013, this workshop has been advised by the committee of discrete mathematics of the Korean Mathematical Society. This year it will take place at The Bloomvista in Yangpyeong on December 20-22, 2021.
The aim of this workshop is to bring together active researchers with different backgrounds to discuss recent and prospective advances in combinatorics and related areas.
We plan to limit the number of participants. Registration (without a fee) will be required. General participants will be required to pay for their own accommodation.

Title: 2021 Combinatorics Workshop (2021 조합론 학술대회)
Date: December 20 – 22 (Mon-Wed), 2021
Venue: The Bloomvista, Yangpyeong
Website: https://cw2021.combinatorics.kr

Plan

Starting in the afternoon of Thursday and ending in the morning of Wednesday.
There will be 5 invited talks for 50 minutes each.
There will be contributed talks for 25 minutes each.

Using English is recommended if there are non-Korean participants in the audience.
The call for contributed talks will be announced later.

It will be an offline meeting.

Invited Speakers

Dongsu Kim, KAIST
Joonkyung Lee, Hanyang University
Hong Liu, University of Warwick, UK
Suil O, SUNY Korea
Seonjeong Park, Jeonju University

Call for Abstracts
We are looking for contributed talks! If you are interested in giving a contributed talk, then please submit your abstract until November 15, 2021.
Registration

Please fill out the registration form by November 21, 2021.

The registration is free.
The organizers will announce the price for the accommodation soon.

Due to the government COVID-19 policy, the organizers may reject some registration, even if the registration was made before November 21.

Organizing Committee

Jeong-Ok Choi, GIST
Sang-il Oum, IBS Discrete Mathematics Group and KAIST
Heesung Shin, Inha University

Advisory Committee
Committee of Discrete Mathematics, The Korean Mathematical Society (Chair: Soojin Cho, Ajou University)
Host and Sponsors
IBS Discrete Mathematics Group

Tuan Tran, Exponential decay of intersection volume with applications on list-decodability and sphere-covering bounds
2021-12-14 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
We give some natural sufficient conditions for balls in a metric space to have small intersection. Roughly speaking, this happens when the metric space is (i) expanding and (ii) well-spread, and (iii) certain random variable on the boundary of a ball has a small tail. As applications, we show that the volume of intersection of balls in Hamming space and symmetric groups decays exponentially as their centers drift apart. To verify condition (iii), we prove some deviation inequalities `on the slice’ for functions with Lipschitz conditions.
We then use these estimates on intersection volumes to

obtain a sharp lower bound on list-decodability of random q-ary codes, confirming a conjecture of Li and Wootters [IEEE Trans. Inf. Theory 2021]; and
improve sphere-covering bound from the 70s on constant weight codes by a factor linear in dimension, resolving a problem raised by Jiang and Vardy [IEEE Trans. Inf. Theory 2004].

Our probabilistic point of view also offers a unified framework to obtain improvements on other sphere-covering bounds, giving conceptually simple and calculation-free proofs for q-ary codes, permutation codes, and spherical codes.
This is joint work with Jaehoon Kim and Hong Liu.

David Munhá Correia, Rainbow matchings
2021-12-09 16:30 - 17:30 | Zoom ID: 869 4632 6610 (ibsdimag)
I will discuss various results for rainbow matching problems. In particular, I will introduce a ‘sampling trick’ which can be used to obtain short proofs of old results as well as to solve asymptotically some well known conjectures. This is joint work with Alexey Pokrovskiy and Benny Sudakov.

Eun-Kyung Cho (조은경), Independent domination of graphs with bounded maximum degree
2021-12-07 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
The independent domination number of a graph $G$, denoted $i(G)$, is the minimum size of an independent dominating set of $G$. In this talk, we prove a series of results regarding independent domination of graphs with bounded maximum degree.
Let $G$ be a graph with maximum degree at most $k$ where $k ge 1$. We prove that if $k = 4$, then $i(G) le frac{5}{9}|V(G)|$, which is tight. Generalizing this result and a result by Akbari et al., we suggest a conjecture on the upper bound of $i(G)$ for $k ge 1$, which is tight if true.
Let $G’$ be a connected $k$-regular graph that is not $K_{k, k}$ where $kgeq 3$. We prove that $i(G’)le frac{k-1}{2k-1}|V(G’)|$, which is tight for $k in {3, 4}$, generalizing a result by Lam, Shiu, and Sun. This result also answers a question by Goddard et al. in the affirmative.
In addition, we show that $frac{i(G’)}{gamma(G’)} le frac{k^3-3k^2+2}{2k^2-6k+2}$, strengthening upon a result of Knor, Škrekovski, and Tepeh, where $gamma(G’)$ is the domination number of $G’$.
Moreover, if we restrict $G’$ to be a cubic graph without $4$-cycles, then we prove that $i(G’) le frac{4}{11}|V(G’)|$, which improves a result by Abrishami and Henning.
This talk is based on joint work with Ilkyoo Choi, Hyemin Kwon, and Boram Park.

Seonghyuk Im (임성혁), Large clique subdivisions in graphs without small dense subgraphs
2021-11-30 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
What is the largest number $f(d)$ where every graph with average degree at least $d$ contains a subdivision of $K_{f(d)}$? Mader asked this question in 1967 and $f(d) = Theta(sqrt{d})$ was proved by Bollobás and Thomason and independently by Komlós and Szemerédi. This is best possible by considering a disjoint union of $K_{d,d}$. However, this example contains a much smaller subgraph with the almost same average degree, for example, one copy of $K_{d,d}$.
In 2017, Liu and Montgomery proposed the study on the parameter $c_{varepsilon}(G)$ which is the order of the smallest subgraph of $G$ with average degree at least $varepsilon d(G)$. In fact, they conjectured that for small enough $varepsilon>0$, every graph $G$ of average degree $d$ contains a clique subdivision of size $Omega(min{d, sqrt{frac{c_{varepsilon}(G)}{log c_{varepsilon}(G)}}})$. We prove that this conjecture holds up to a multiplicative $min{(loglog d)^6,(log log c_{varepsilon}(G))^6}$-term.
As a corollary, for every graph $F$, we determine the minimum size of the largest clique subdivision in $F$-free graphs with average degree $d$ up to multiplicative polylog$(d)$-term.
This is joint work with Jaehoon Kim, Youngjin Kim, and Hong Liu.

Tuukka Korhonen, Fast FPT-Approximation of Branchwidth
2021-11-25 16:30 - 17:30 | Zoom ID: 869 4632 6610 (ibsdimag)
Branchwidth determines how graphs, and more generally, arbitrary connectivity (basically symmetric and submodular) functions could be decomposed into a tree-like structure by specific cuts. We develop a general framework for designing fixed-parameter tractable (FPT) 2-approximation algorithms for branchwidth of connectivity functions. The first ingredient of our framework is combinatorial. We prove a structural theorem establishing that either a sequence of particular refinement operations could decrease the width of a branch decomposition or that the width of the decomposition is already within a factor of 2 from the optimum. The second ingredient is an efficient implementation of the refinement operations for branch decompositions that support efficient dynamic programming. We present two concrete applications of our general framework.

An algorithm that for a given n-vertex graph G and integer k in time $2^{2^{O(k)}} n^2$ either constructs a rank decomposition of G of width at most 2k or concludes that the rankwidth of G is more than $k$. It also yields a $(2^{2k+1}−1)$-approximation algorithm for cliquewidth within the same time complexity, which in turn, improves to $f(k) n^2$ the running times of various algorithms on graphs of cliquewidth $k$. Breaking the “cubic barrier” for rankwidth and cliquewidth was an open problem in the area.
An algorithm that for a given n-vertex graph G and integer k in time $2^{O(k)} n$ either constructs a branch decomposition of G of width at most $2k$ or concludes that the branchwidth of G is more than $k$. This improves over the 3-approximation that follows from the recent treewidth 2-approximation of Korhonen [FOCS 2021].

This is joint work with Fedor Fomin.

Casey Tompkins, Ramsey numbers of Boolean lattices
2021-11-23 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
The poset Ramsey number $R(Q_{m},Q_{n})$ is the smallest integer $N$ such that any blue-red coloring of the elements of the Boolean lattice $Q_{N}$ has a blue induced copy of $Q_{m}$ or a red induced copy of $Q_{n}$. Axenovich and Walzer showed that $n+2le R(Q_{2},Q_{n})le2n+2$. Recently, Lu and Thompson
improved the upper bound to $frac{5}{3}n+2$. In this paper, we solve this problem asymptotically by showing that $R(Q_{2},Q_{n})=n+O(n/log n)$.
Joint work with Dániel Grósz and Abhishek Methuku.

Graph Product Structure Theory: Gathering of Participants from Korea
2021-11-23 - 2021-11-26 | Room B223, IBS (기초과학연구원)
On November 22-26, 2021, there is a “Graph Product Structure Theory” workshop in BIRS Centre in Banff (https://www.birs.ca/events/2021/5-day-workshops/21w5235), organized in a hybrid manner with 15 onsite participants and around 50 remote participants. We would like to meet in a group of 5-10 remote participants from Korea in one place, creating a secondary workshop site in Korea. This would allow joint participation in online talks and in-person discussions between local participants.

Sebastian Wiederrecht, Matching Minors in Bipartite Graphs
2021-11-11 16:30 - 17:30 | Zoom ID: 869 4632 6610 (ibsdimag)
Matching minors are a specialisation of minors which preserves the existence and elementary structural properties of perfect matchings. They were first discovered as part of the study of the Pfaffian recognition problem on bipartite graphs (Polya’s Permanent Problem) and acted as a major inspiration for the definition of butterfly minors in digraphs. In this talk we consider the origin and motivation behind the study of matching minors, the current state of the art, and their relation to structural digraph theory. The main result is a generalisation of the structure theorem by Robertson et al. and McCuaig for $K_{3,3}$-matching minor free bipartite graphs to bipartite graphs excluding $K_{t,t}$ as a matching minor for general t. This generalisation can be seen as a matching theoretic version of the Flat Wall Theorem by Robertson and Seymour.

Jaehoon Kim (김재훈), 2-complexes with unique embeddings in 3-space
2021-11-09 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
A well-known theorem of Whitney states that a 3-connected planar graph admits an essentially unique embedding into the 2-sphere. We prove a 3-dimensional analogue: a simply-connected 2-complex every link graph of which is 3-connected admits an essentially unique locally flat embedding into the 3-sphere, if it admits one at all. This can be thought of as a generalisation of the 3-dimensional Schoenflies theorem. This is joint work with Agelos Georgakopoulos.

Martin Milanič, Tree Decompositions with Bounded Independence Number
2021-11-05 16:30 - 17:30 | Zoom ID: 869 4632 6610 (ibsdimag)
The independence number of a tree decomposition $mathcal{T}$ of a graph is the smallest integer $k$ such that each bag of $mathcal{T}$ induces a subgraph with independence number at most $k$. If a graph $G$ is given together with a tree decomposition with bounded independence number, then the Maximum Weight Independent Set (MWIS) problem can be solved in polynomial time. Motivated by this observation, we consider six graph containment relations—the subgraph, topological minor, and minor relations, as well as their induced variants—and for each of them characterize the graphs $H$ for which any graph excluding $H$ with respect to the relation admits a tree decomposition with bounded independence number. Furthermore, using a variety of tools including SPQR trees and potential maximal cliques, we show how to obtain such tree decompositions efficiently.
As an immediate consequence, we obtain that the MWIS problem can be solved in polynomial time in an infinite family of graph classes that properly contain the class of chordal graphs. In fact, our approach shows that the Maximum Weight Independent $mathcal{H}$-Packing problem, a common generalization of the MWIS and the Maximum Weight Induced Matching problems, can be solved in polynomial time in these graph classes.
This is joint work with Clément Dallard and Kenny Štorgel.

Ben Lund, Maximal 3-wise intersecting families
2021-11-02 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
A family $mathcal F$ of subsets of {1,2,…,n} is called maximal k-wise intersecting if every collection of at most k members from $mathcal F$ has a common element, and moreover, no set can be added to $mathcal F$ while preserving this property. In 1974, Erdős and Kleitman asked for the smallest possible size of a maximal k-wise intersecting family, for k≥3. We resolve this problem for k=3 and n even and sufficiently large.
This is joint work with Kevin Hendrey, Casey Tompkins, and Tuan Tran.

Donggyu Kim (김동규), 𝝘-graphic delta-matroids and their applications
2021-10-26 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
Bouchet (1987) defined delta-matroids by relaxing the base exchange axiom of matroids. Oum (2009) introduced a graphic delta-matroid from a pair of a graph and its vertex subset. We define a $Gamma$-graphic delta-matroid for an abelian group $Gamma$, which generalizes a graphic delta-matroid.
For an abelian group $Gamma$, a $Gamma$-labelled graph is a graph whose vertices are labelled by elements of $Gamma$. We prove that a certain collection of edge sets of a $Gamma$-labelled graph forms a delta-matroid, which we call a $Gamma$-graphic delta-matroid, and provide a polynomial-time algorithm to solve the separation problem, which allows us to apply the symmetric greedy algorithm of Bouchet (1987) to find a maximum weight feasible set in such a delta-matroid. We also prove that a $Gamma$-graphic delta-matroid is a graphic delta-matroid if and only if it is even. We prove that every $mathbb{Z}_p^k$-graphic delta matroid is represented by some symmetric matrix over a field of characteristic of order $p^k$, and if every $Gamma$-graphic delta-matroid is representable over a finite field $mathbb{F}$, then $Gamma$ is isomorphic to $mathbb{Z}_p^k$ and $mathbb{F}$ is a field of order $p^ell$ for some prime $p$ and positive integers $k$ and $ell$.
This is joint work with Duksang Lee and Sang-il Oum.

Young Researchers in Extremal and Probabilistic Combinatorics
2021-10-20 - 2021-10-22 | Zoom ID: 554 788 7710 (507464)
The aim of the Young Researchers in Extremal and Probabilistic Combinatorics is to bring together early career researchers working on these topics. The workshop will consist of several 25 minute talks across three days from October 20 to 22, 2021. Due to Covid the workshop will be held online.

Invited Speakers & Program
Oct. 20 Wednesday 4 PM (KST) – 8 PM (KST)

Andrzej Grzesik (Jagiellonian University): Degenerated generalized Turán numbers of cycles, 4:00-4:25
Jan Volec (Czech Technical University): Existence of common graphs with large chromatic number, 4:30-4:55
Balázs Patkós (Rényi Institute): Vector sum-intersection theorems, 5:00-5:25
István Tomon (ETH Zurich): Small doubling, atomic structure and $ell$-divisible set families, 5:30-5:55
Michael Anastos (Freie Universität Berlin): Longest Cycles in Sparse Random Graphs and Where to Find Them, 6:30-6:55
Adam Zsolt Wagner (Tel Aviv University): Constructions in combinatorics via neural networks, 7:00-7:25
Yelena Yuditsky (Université libre de Bruxelles): On multicolor Ramsey numbers and subset-coloring of hypergraphs, 7:30-7:55

Oct. 21 Thursday 4 PM (KST) – 8 PM (KST)

Kevin Hendrey (IBS Discrete Mathematics Group): Extremal functions for sparse minors, 4:00-4:25
Simona Boyadzhiyska (Freie Universität Berlin): Ramsey simplicity of random graphs, 4:30-4:55
Shagnik Das (National Taiwan University): Schur’s Theorem in randomly perturbed sets, 5:00-5:25
Tony Huynh (Monash University): Subgraph densities in minor-closed classes (and beyond), 5:30-5:55
Ander Lamaison (Masaryk University): Hypergraphs with minimum uniform Turán density, 6:30-6:55
Ben Lund (IBS Discrete Mathematics Group): Maximal 3-wise intersecting families, 7:00-7:25
Liana Yepremyan (London School of Economics): Enumerating independent sets in Abelian Cayley graphs, 7:30-7:55

Oct. 22 Friday 4 PM (KST) – 8 PM (KST)

Andrey Kupavskii (CNRS): Binary scalar products, 4:00-4:25
Dániel Gerbner (Rényi Institute): Exact results for generalized Turán problems, 4:30-4:55
Oliver Janzer (University of Cambridge): Tiling with monochromatic bipartite graphs of bounded maximum degree, 5:00-5:25
Nika Salia (Rényi Institute): Pósa-type results for Berge Hypergraphs, 5:30-5:55
Debsoumya Chakraborti (IBS Discrete Mathematics Group): Mixing time and expanders, 6:30-6:55
Wei-Tian Li (National Chung Hsing University): Ramsey Properties for $V$-shaped Posets in the Boolean Lattices, 7:00-7:25
Dimitry Zakharov (Moscow Institute of Physics and Technology): Zero subsums in vector spaces over finite fields, 7:30-7:55

Abstracts (pdf file)

Joonkyung Lee (이준경), Majority dynamics on sparse random graphs
2021-10-12 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
Majority dynamics on a graph $G$ is a deterministic process such that every vertex updates its $pm 1$-assignment according to the majority assignment on its neighbor simultaneously at each step. Benjamini, Chan, O’Donnell, Tamuz and Tan conjectured that, in the Erdős-Rényi random graph $G(n,p)$, the random initial $pm 1$-assignment converges to a $99%$-agreement with high probability whenever $p=omega(1/n)$.
This conjecture was first confirmed for $pgeqlambda n^{-1/2}$ for a large constant $lambda$ by Fountoulakis, Kang and Makai. Although this result has been reproved recently by Tran and Vu and by Berkowitz and Devlin, it was unknown whether the conjecture holds for $p< lambda n^{-1/2}$. We break this $Omega(n^{-1/2})$-barrier by proving the conjecture for sparser random graphs $G(n,p)$, where $lambda’ n^{-3/5}log n leq p leq lambda n^{-1/2}$ with a large constant $lambda’>0$.
Joint work with Debsoumya Chakraborti, Jeong Han Kim and Tuan Tran.

Paul Seymour, Polynomial bounds for chromatic number
2021-10-08 10:00 - 11:00 | Zoom ID: 869 4632 6610 (ibsdimag)
The Gyárfás-Sumner conjecture says that for every forest $H$, there is a function $f$ such that the chromatic number $chi(G)$ is at most $f(omega(G))$ for every $H$-free graph $G$ (“$H$-free” means with no induced subgraph isomorphic to $H$, and $omega(G)$ is the size of the largest clique of $G$). This well-known conjecture has been proved only for a few types of forest.
Nevertheless, there is a much stronger conjecture, due to Esperet: that for every forest $H$, there is a polynomial function $f$ as above. As one might expect, this has been proved for even fewer types of forest; and the smallest tree $H$ for which Esperet’s conjecture is not known is the five-vertex path $P_5$.
A third notorious conjecture is the Erdős-Hajnal conjecture, that for every graph $H$, there exists $c>0$ such that $alpha(G)omega(G)ge |G|^c$ for every $H$-free graph $G$ (where $alpha(G)$ is the size of the largest stable set of $G$). The smallest graph $H$ for which this is not known is also $P_5$, which, conveniently, is a forest; and every forest that satisfies Esperet’s conjecture also satisfies the Erdős-Hajnal conjecture. So there is substantial interest in the chromatic numbers of $P_5$-free graphs. The best upper bound that was previously known, due to Esperet, Lemoine, Maffray, and Morel, was that $chi(G)le (5/27)3^omega(G)$ for every $P_5$-free graph $G$ with $omega(G) > 2$. In recent work with Alex Scott and Sophie Spirkl, we have proved several results related to Esperet’s conjecture, including proofs of its truth for some new types of forest $H$, and a “near-polynomial” bound when $H = P_5$, that $chi(G) le omega(G)^{log_2(omega(G))}$ for every $P_5$-free graph $G$ with $omega(G) > 2$. We survey these results and give a proof of the new bound for $P_5$.

Eunjin Oh (오은진), Feedback Vertex Set on Geometric Intersection Graphs
2021-10-05 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
I am going to present an algorithm for computing a feedback vertex set of a unit disk graph of size k, if it exists, which runs in time $2^{O(sqrt{k})}(n + m)$, where $n$ and $m$ denote the numbers of vertices and edges, respectively. This improves the $2^{O(sqrt{k}log k)}(n + m)$-time algorithm for this problem on unit disk graphs by Fomin et al. [ICALP 2017].

Péter Pál Pach, The Alon-Jaeger-Tarsi conjecture via group ring identities
2021-09-30 16:30 - 17:30 | Zoom ID: 869 4632 6610 (ibsdimag)
The Alon-Jaeger-Tarsi conjecture states that for any finite field $mathbb{F}$ of size at least 4 and any nonsingular matrix $M$ over $mathbb{F}$ there exists a vector $x$ such that neither $x$ nor $Mx$ has a 0 component. In joint work with János Nagy we proved this conjecture when the size of the field is sufficiently large, namely, when $61

Kevin Hendrey, Extremal functions for sparse minors
2021-09-28 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
The extremal function $c(H)$ of a graph $H$ is the supremum of densities of graphs not containing $H$ as a minor, where the density of a graph is the ratio of the number of edges to the number of vertices. Myers and Thomason (2005), Norin, Reed, Thomason and Wood (2020), and Thomason and Wales (2019) determined the asymptotic behaviour of $c(H)$ for all polynomially dense graphs $H$, as well as almost all graphs of constant density. We explore the asymptotic behavior of the extremal function in the regime not covered by the above results, where in addition to having constant density the graph $H$ is in a graph class admitting strongly sublinear separators. We establish asymptotically tight bounds in many cases. For example, we prove that for every planar graph $H$, [c(H) = (1+o(1))max (v(H)/2, v(H)-alpha(H)),] extending recent results of Haslegrave, Kim and Liu (2020). Joint work with Sergey Norin and David R. Wood.

Dabeen Lee (이다빈), Mixing sets, submodularity, and chance-constrained optimization
2021-09-07 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
A particularly important substructure in modeling joint linear chance-constrained programs with random right-hand sides and finite sample space is the intersection of mixing sets with common binary variables (and possibly a knapsack constraint). In this talk, we first explain basic mixing sets by establishing a strong and previously unrecognized connection to submodularity. In particular, we show that mixing inequalities with binary variables are nothing but the polymatroid inequalities associated with a specific submodular function. This submodularity viewpoint enables us to unify and extend existing results on valid inequalities and convex hulls of the intersection of multiple mixing sets with common binary variables. Then, we study such intersections under an additional linking constraint lower bounding a linear function of the continuous variables. This is motivated from the desire to exploit the information encoded in the knapsack constraint arising in joint linear CCPs via the quantile cuts. We propose a new class of valid inequalities and characterize when this new class along with the mixing inequalities are sufficient to describe the convex hull. This is based on joint work with Fatma Fatma Kılınç-Karzan and Simge Küçükyavuz.

Cheolwon Heo (허철원), Representations of even-cycle matroids
2021-08-31 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
A signed graph is a pair $(G,Sigma)$ where $G$ is a graph and $Sigma$ is a subset of edges of $G$. A cycle $C$ of $G$ is a subset of edges of $G$ such that every vertex of the subgraph of $G$ induced by $C$ has an even degree. We say that $C$ is even in $(G,Sigma)$ if $|C cap Sigma|$ is even; otherwise, $C$ is odd. A matroid $M$ is an even-cycle matroid if there exists a signed graph $(G,Sigma)$ such that circuits of $M$ precisely corresponds to inclusion-wise minimal non-empty even cycles of $(G,Sigma)$. For even-cycle matroids, two fundamental questions arise:
(1) what is the relationship between two signed graphs representing the same even-cycle matroids?
(2) how many signed graphs can an even-cycle matroid have?
For (a), we characterize two signed graphs $(G_1,Sigma_1)$ and $(G_2,Sigma_2)$ where $G_1$ and $G_2$ are $4$-connected that represent the same even-cycle matroids.
For (b), we introduce pinch-graphic matroids, which can generate exponentially many representations even when the matroid is $3$-connected. An even-cycle matroid is a pinch-graphic matroid if there exists a signed graph with a pair of vertices such that every odd cycle intersects with at least one of them. We prove that there exists a constant $c$ such that if a matroid is even-cycle matroid that is not pinch-graphic, then the number of representations is bounded by $c$. This is joint work with Bertrand Guenin and Irene Pivotto.

Eun Jung Kim (김은정), A Constant-factor Approximation for Weighted Bond Cover
2021-08-24 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
The Weighted $mathcal F$-Vertex Deletion for a class $mathcal F$ of graphs asks, given a weighted graph $G$, for a minimum weight vertex set $S$ such that $G-Sinmathcal F$. The case when $mathcal F$ is minor-closed and excludes some graph as a minor has received particular attention but a constant-factor approximation remained elusive for Weighted $mathcal F$-Vertex Deletion. Only three cases of minor-closed $mathcal F$ are known to admit constant-factor approximations, namely Vertex Cover, Feedback Vertex Set and Diamond Hitting Set.
We study the problem for the class $mathcal F$ of $theta_c$-minor-free graphs, under the equivalent setting of the Weighted c-Bond Cover, and present a constant-factor approximation algorithm using the primal-dual method. For this, we leverage a structure theorem implicit in [Joret et al., SIDMA’14] which states the following: any graph $G$ containing a $theta_c$-minor-model either contains a large two-terminal protrusion, or contains a constant-size $theta_c$-minor-model, or a collection of pairwise disjoint constant-sized connected sets that can be contracted simultaneously to yield a dense graph. In the first case, we tame the graph by replacing the protrusion with a special-purpose weighted gadget. For the second and third case, we provide a weighting scheme which guarantees a local approximation ratio. Besides making an important step in the quest of (dis)proving a constant-factor approximation for Weighted $mathcal F$-Vertex Deletion, our result may be useful as a template for algorithms for other minor-closed families.
This is joint work with Euiwoong Lee and Dimitrios M. Thilikos.

Petr Hliněný, Twin-width is linear in the poset width
2021-08-18 17:00 - 18:00 | Zoom ID: 869 4632 6610 (ibsdimag)
Twin-width is a new parameter informally measuring how diverse are the neighbourhoods of the graph vertices, and it extends also to other binary relational structures, e.g. to digraphs and posets. It was introduced quite recently, in 2020 by Bonnet, Kim, Thomassé, and Watrigant. One of the core results of these authors is that FO model checking on graph classes of bounded twin-width is in FPT. With that result, they also claimed that posets of bounded width have bounded twin-width, thus capturing a prior result on FO model checking of posets of bounded width in FPT. However, their translation from poset width to twin-width was indirect and giving only a very loose double-exponential bound.
We prove that posets of width d have twin-width at most 9d with a direct and elementary argument, and show that this bound is tight up to a constant factor. Specifically, for posets of width 2, we prove that in the worst case their twin-width is also equal to 2. These two theoretical results are complemented with straightforward algorithms to construct the respective contraction sequence for a given poset.
(Joint work with my student Jakub Balaban who obtained the main ideas in his bachelor thesis.)

Linda Cook, Two results on graphs with holes of restricted lengths
2021-08-17 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
We call an induced cycle of length at least four a hole. The parity of a hole is the parity of its length. Forbidding holes of certain types in a graph has deep structural implications. In 2006, Chudnovksy, Seymour, Robertson, and Thomas famously proved that a graph is perfect if and only if it does not contain an odd hole or a complement of an odd hole. In 2002, Conforti, Cornuéjols, Kapoor, and Vuškovíc provided a structural description of the class of even-hole-free graphs. I will describe the structure of all graphs that contain only holes of length $ell$ for every $ell geq 7$ (joint work with Jake Horsfield, Myriam Preissmann, Paul Seymour, Ni Luh Dewi Sintiari, Cléophée Robin, Nicolas Trotignon, and Kristina Vuškovíc.
Analysis of how holes interact with graph structure has yielded detection algorithms for holes of various lengths and parities. In 1991, Bienstock showed it is NP-Hard to test whether a graph G has an even (or odd) hole containing a specified vertex $v in V(G)$. In 2002, Conforti, Cornuéjols, Kapoor, and Vuškovíc gave a polynomial-time algorithm to recognize even-hole-free graphs using their structure theorem. In 2003, Chudnovsky, Kawarabayashi, and Seymour provided a simpler and slightly faster algorithm to test whether a graph contains an even hole. In 2019, Chudnovsky, Scott, Seymour, and Spirkl provided a polynomial-time algorithm to test whether a graph contains an odd hole. Later that year, Chudnovsky, Scott, and Seymour strengthened this result by providing a polynomial-time algorithm to test whether a graph contains an odd hole of length at least $ell$ for any fixed integer $ell geq 5$. I will present a polynomial-time algorithm (joint work with Paul Seymour) to test whether a graph contains an even hole of length at least $ell$ for any fixed integer $ell geq 4$.

Duksang Lee (이덕상), Intertwining connectivities for vertex-minors and pivot-minors
2021-08-10 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
We show that for pairs (Q,R) and (S,T) of disjoint subsets of vertices of a graph G, if G is sufficiently large, then there exists a vertex v in V(G)−(Q∪R∪S∪T) such that there are two ways to reduce G by a vertex-minor operation while preserving the connectivity between Q and R and the connectivity between S and T. Our theorem implies an analogous theorem of Chen and Whittle (2014) for matroids restricted to binary matroids. Joint work with Sang-il Oum.

Maria Chudnovsky, Induced subgraphs and tree decompositions
2021-07-28 15:00 - 16:00 | Zoom ID: 869 4632 6610 (ibsdimag)
Tree decompositions are a powerful tool in structural graph theory; they are traditionally used in the context of forbidden graph minors. Connecting tree decompositions and forbidden induced subgraphs has until recently remained out of reach.
Tree decompositions are closely related to the existence of “laminar collections of separations” in a graph, which roughly means that the separations in the collection “cooperate” with each other, and the pieces that are obtained when the graph is simultaneously decomposed by all the separations in the collection “line up” to form a tree structure. Such collections of separations come up naturally in the context of forbidden minors.
In the case of families where induced subgraphs are excluded, while there are often natural separations, they are usually very far from forming a laminar collection. In what follows we mostly focus on families of graphs of bounded degree. It turns out that due to the bound on the degree, these collections of natural separations can be partitioned into a bounded number of laminar collections. This in turn allows to us obtain a wide variety of structural and algorithmic results, which we will survey in this talk.

Euiwoong Lee (이의웅), The Karger-Stein algorithm is optimal for k-cut
2021-07-27 15:00 - 16:00 | Room B232, IBS (기초과학연구원)
In the k-cut problem, we are given an edge-weighted graph and want to find the least-weight set of edges whose deletion breaks the graph into k connected components. It is easy to see that the elegant randomized contraction algorithm of Karger and Stein for global mincut (k=2) can be naturally extended for general k with the running time of $O(n^{2k-2})$. Using tree packings, Thorup gave a deterministic algorithm that has the same running time.
In this work, we show that for any fixed $kge 2$, the Karger-Stein algorithm outputs any fixed minimum k-cut with probability at least $Omega(n^{-k})$. This also gives an extremal bound of $O(n^k)$ on the number of minimum k-cuts in an n-vertex graph and an algorithm to compute a minimum k-cut in similar runtime. Both are essentially tight. The first main ingredient in our result is a fine-grained analysis of how the graph shrinks—and how the average degree evolves—under the Karger-Stein process. The second ingredient is an extremal result bounding the number of cuts of size at most $(2-delta)OPT/k$, using the Sunflower lemma.
Joint work with Anupam Gupta, David G. Harris, and Jason Li.
