Discrete Mathematics Group

Pascal Gollin, A Cantor-Bernstein-type theorem for spanning trees in infinite graphs
2019-10-29 16:30 - 17:30 (Asia/Seoul) | Room B232, IBS (기초과학연구원)
Given a cardinal $lambda$, a $lambda$-packing of a graph $G$ is a family of $lambda$ many edge-disjoint spanning trees of $G$, and a $lambda$-covering of $G$ is a family of spanning trees covering $E(G)$.
We show that if a graph admits a $lambda$-packing and a $lambda$-covering then the graph also admits a decomposition into $lambda$ many spanning trees. In this talk, we concentrate on the case of $lambda$ being an infinite cardinal. Moreover, we will provide a new and simple proof for a theorem of Laviolette characterising the existence of a $lambda$-packing, as well as for a theorem of Erdős and Hajnal characterising the existence of a $lambda$-covering.
Joint work with Joshua Erde, Attila Joó, Paul Knappe and Max Pitz.

Extremal and Structural Graph Theory (2019 KMS Annual Meeting)
2019-10-26 - 2019-10-27 | Room 426, Hong-Mun Hall, Hongik University, Seoul
Focus Session @ 2019 KMS Annual Meeting
A focus session “Extremal and Structural Graph Theory” at the 2019 KMS Annual Meeting is organized by Sang-il Oum. URL: http://www.kms.or.kr/meetings/fall2019/
Speakers

Ilkyoo Choi (최일규), Hankuk University of Foreign Studies
Kevin Hendrey, IBS Discrete Mathematics Group
Pascal Gollin, IBS Discrete Mathematics Group
Jaehoon Kim (김재훈), KAIST
Ringi Kim (김린기), KAIST
Seog-Jin Kim (김석진), Konkuk University
O-joung Kwon (권오정), Incheon National University / IBS Discrete Mathematics Group
Zi-Xia Song (宋梓霞), University of Central Florida
Casey Tompkins, IBS Discrete Mathematics Group

Schedules

Slot A: October 26, 2019 Saturday, 9:00am-10:15am

Seog-Jin Kim
Kevin Hendrey
Ilkyoo Choi

Slot B: October 26, 2019 Saturday, 10:40am-11:55pm

Jaehoon Kim
Ringi Kim
Casey Tompkins

Slot D: October 27, 2019 Sunday, 9:00am-10:15am

Zi-Xia Song
Pascal Gollin
O-joung Kwon

Abstracts
Ilkyoo Choi (최일규), Degeneracy and colorings of squares of planar graphs without 4-cycles
We prove several results on coloring squares of planar graphs without 4-cycles. First, we show that if $G$ is such a graph, then $G^2$ is $(Delta(G)+72)$-degenerate. This implies an upper bound of $Delta(G)+73$ on the chromatic number of $G^2$ as well as on several variants of the chromatic number such as the list-chromatic number, paint number, Alon–Tarsi number, and correspondence chromatic number. We also show that if $Delta(G)$ is sufficiently large, then the upper bounds on each of these parameters of $G^2$ can all be lowered to $Delta(G)+2$ (which is best possible). To complement these results, we show that 4-cycles are unique in having this property. Specifically, let $S$ be a finite list of positive integers, with $4
otin S$. For each constant $C$, we construct a planar graph $G_{S,C}$ with no cycle with length in $S$, but for which $chi(G_{S,C}^2) > Delta(G_{S,C})+C$. This is joint work with Dan Cranston and Theo Pierron.
Kevin Hendrey, Defective and clustered choosability of sparse graphs
An (improper) graph colouring has defect $d$ if each monochromatic subgraph has maximum degree at most $d$, and has clustering $c$ if each monochromatic component has at most $c$ vertices. Given an infinite class of graphs $mathcal{G}$, it is interesting to ask for the minimum integer $chi_{Delta}(mathcal{G})$ for which there exist an integer $d$ such that every graph in $mathcal{G}$ has a $chi_{Delta}(mathcal{G})$-colouring with defect at most $d$, and the minimum integer $chi_{star}(mathcal{G})$-colouring for which there exist an integer $c$ such that every graph in $mathcal{G}$ has a $chi_{star}(mathcal{G})$-colouring with clustering at most $c$. We explore clustered and defective colouring in graph classes with bounded maximum average degree. As an example, our results show that every earth-moon graph has an 8-colouring with clustering at most 405. Our results hold in the stronger setting of list-colouring. Joint work with David Wood.
Pascal Gollin, Progress on the Ubiquity Conjecture
A classic result of Halin says that if a graph $G$ contains for every $n in mathbb{N}$ as subgraphs $n$ disjoint rays, i.e., one-way infinite paths, then $G$ already contains infinitely many disjoint rays as subgraphs. We say a graph $G$ is ubiquitous with respect to the subgraph relation if whenever a graph $Gamma$ contains $n$ disjoint copies of $G$ as a subgraph for all $n in mathbb{N}$, then $Gamma$ already contains infinitely many disjoint copies of $G$ as a subgraph. We define ubiquity w.r.t. the minor relation or topological minor relation analogously. A fundamental conjecture about infinite graphs due to Andreae is the Ubiquity Conjecture. It states that every locally finite connected graph is ubiquitous w.r.t. the minor relation. In a series of papers we make progress on the conjecture proving various ubiquity results w.r.t. both the topological minor relation and the minor relation, making use of well-quasi ordering techniques.
Jaehoon Kim (김재훈), A rainbow version of Dirac’s theorem
For a collection $mathbf{G}={G_1,dots, G_s}$ of graphs, we say that a graph $H$ is $mathbf{G}$-transversal, or $mathbf{G}$-rainbow, if there exists a bijection $phi:E(H)rightarrow [s]$ with $ein G_{phi(e)}$ for all $ein E(H)$. Aharoni conjectured that if for each $iin [r]$, then graph $G_i$ is an $n$-vertex graph on the same vertex set $V$ and $delta(G_i)geq n/2$ for all $iin [s]$, then there exists a $mathbf{G}$-transversal Hamilton cycle on $V$. We prove this conjecture. We also prove a similar result for $K_r$-factors. This is joint work with Felix Joos.
Ringi Kim (김린기), Obstructions for partitioning into forests
For a class $mathcal{C}$ of graphs, we de fine $mathcal{C}$-edge-brittleness of a graph $G$ as the minimum $ell$ such that the vertex set of $G$ can be partitioned into sets inducing a subgraph in $mathcal{C}$ and there are $ell$ edges having ends in distinct parts. In this talk, we characterize classes of graphs having bounded $mathcal{C}$-edge-brittleness for a class $mathcal{C}$ of forests in terms of forbidden obstructions. This is joint work with Sang-il Oum and Sergey Norin.
Seog-Jin Kim (김석진), The Alon-Tarsi number of subgraphs of a planar graph
This paper constructs a planar graph $G_1$ such that for any subgraph $H$ of $G_1$ with maximum degree $Delta(H) le 3$, $G_1-E(H)$ is not $3$-choosable, and a planar graph $G_2$ such that for any star forest $F$ in $G_2$, $G_2-E(F)$ contains a copy of $K_4$ and hence $G_2-E(F)$ is not $3$-colourable. On the other hand, we prove that every planar graph $G$ contains a forest $F$ such that the Alon-Tarsi number of $G – E(F)$ is at most $3$, and hence $G – E(F)$ is 3-paintable and 3-choosable. This is joint work with Ringi Kim and Xuding Zhu.
O-joung Kwon (권오정), Erdős-Pósa property of H-induced subdivisions
A class $mathcal{F}$ of graphs has the induced Erdős-Pósa property if there exists a function $f$ such that for every graph $G$ and every positive integer $k$, $G$ contains either $k$ pairwise vertex-disjoint induced subgraphs that belong to $mathcal{F}$, or a vertex set of size at most $f(k)$ hitting all induced copies of graphs in $mathcal{F}$. Kim and Kwon (SODA’18) showed that for a cycle $C_{ell}$ of length $ell$, the class of $C_{ell}$-subdivisions has the induced Erdős-Pósa property if and only if $ellle 4$. In this paper, we investigate whether or not the class of $H$-subdivisions has the induced Erdős-Pósa property for other graphs $H$. We completely settle the case when $H$ is a forest or a complete bipartite graph. Regarding the general case, we identify necessary conditions on $H$ for the class of $H$-subdivisions to have the induced Erdős-Pósa property. For this, we provide three basic constructions that are useful to prove that the class of the subdivisions of a graph does not have the induced Erdős-Pósa property. Among remaining graphs, we prove that if $H$ is either the diamond, the $1$-pan, or the $2$-pan, then the class of $H$-subdivisions has the induced Erdős-Pósa property.
Zi-Xia Song, On the size of $(K_t,mathcal{T}_k)$-co-critical graphs
Given an integer $rge1$ and graphs $G, H_1, ldots, H_r$, we write $G rightarrow ({H}_1, ldots, {H}_r)$ if every $r$-coloring of the edges of $G$ contains a monochromatic copy of $H_i$ in color $i$ for some $iin{1, ldots, r}$. A non-complete graph $G$ is $(H_1, ldots, H_r)$-co-critical if $G
rightarrow ({H}_1, ldots, {H}_r)$, but $G+erightarrow ({H}_1, ldots, {H}_r)$ for every edge $e$ in $overline{G}$. Motivated by Hanson and Toft’s conjecture, we study the minimum number of edges over all $(K_t, mathcal{T}_k)$-co-critical graphs on $n$ vertices, where $mathcal{T}_k$ denotes the family of all trees on $k$ vertices. In this talk, we will present our recent results on this topic. This is joint work with Jingmei Zhang.
Casey Tompkins, Generalizations of the Erdős-Gallai theorem
A classical result of Erdős and Gallai bounds the number of edges in an $n$-vertex graph with no path of length $k$ (denoted $P_k$). In this talk, I will discuss generalizations of this result in multiple settings. In one direction, I will consider so-called generalized Turán problems for $P_k$-free graphs. That is, rather than maximizing the number of edges, we consider maximizing the number of copies of some graph $H$. Building on results of Luo where $H$ is a clique, we consider the case when $H$ is also a path. In another direction, I will consider Erdős-Gallai type problems in a hypergraph setting. Here I will discuss recent results involving forbidding Berge copies of a path, including the solution to some problems and conjectures of Győri, Katona and Lemons as well Füredi, Kostochka and Luo. I will also mention some Kopylov-type variants of this problem where the hypergraph is assumed to be connected. Moreover, I will discuss some recent work on forbidding Berge copies of a tree. Finally, as time permits I will mention some colored variants of the Erdős-Gallai problem. The new results presented are joint work with various subsets of the authors Akbar Davoodi, Beka Ergemlidze, Ervin Győri, Abhishek Methuku, Nika Salia, Mate Vizer, Oscar Zamora.

Joonkyung Lee (이준경), On some properties of graph norms
2019-10-22 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
For a graph $H$, its homomorphism density in graphs naturally extends to the space of two-variable symmetric functions $W$ in $L^p$, $pgeq e(H)$, denoted by $t_H(W)$. One may then define corresponding functionals $|W|_{H}:=|t_H(W)|^{1/e(H)}$ and $|W|_{r(H)}:=t_H(|W|)^{1/e(H)}$ and say that $H$ is (semi-)norming if $|.|_{H}$ is a (semi-)norm and that $H$ is weakly norming if $|.|_{r(H)}$ is a norm.
We obtain some results that contribute to the theory of (weakly) norming graphs. Firstly, we show that ‘twisted’ blow-ups of cycles, which include $K_{5,5}setminus C_{10}$ and $C_6square K_2$, are not weakly norming. This answers two questions of Hatami, who asked whether the two graphs are weakly norming. Secondly, we prove that $|.|_{r(H)}$ is not uniformly convex nor uniformly smooth, provided that $H$ is weakly norming. This answers another question of Hatami, who estimated the modulus of convexity and smoothness of $|.|_{H}$. We also prove that every graph $H$ without isolated vertices is (weakly) norming if and only if each component is an isomorphic copy of a (weakly) norming graph. This strong factorisation result allows us to assume connectivity of $H$ when studying graph norms. Based on joint work with Frederik Garbe, Jan Hladký, and Bjarne Schülke.

Zi-Xia Song (宋梓霞), Ramsey numbers of cycles under Gallai colorings
2019-10-15 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
For a graph $H$ and an integer $kge1$, the $k$-color Ramsey number $R_k(H)$ is the least integer $N$ such that every $k$-coloring of the edges of the complete graph $K_N$ contains a monochromatic copy of $H$. Let $C_m$ denote the cycle on $mge4 $ vertices. For odd cycles, Bondy and ErdH{o}s in 1973 conjectured that for all $kge1$ and $nge2$, $R_k(C_{2n+1})=ncdot 2^k+1$. Recently, this conjecture has been verified to be true for all fixed $k$ and all $n$ sufficiently large by Jenssen and Skokan; and false for all fixed $n$ and all $k$ sufficiently large by Day and Johnson. Even cycles behave rather differently in this context. Little is known about the behavior of $R_k(C_{2n})$ in general. In this talk we will present our recent results on Ramsey numbers of cycles under Gallai colorings, where a Gallai coloring is a coloring of the edges of a complete graph without rainbow triangles. We prove that the aforementioned conjecture holds for all $k$ and all $n$ under Gallai colorings. We also completely determine the Ramsey number of even cycles under Gallai colorings.
Joint work with Dylan Bruce, Christian Bosse, Yaojun Chen and Fangfang Zhang.

Alexandr V. Kostochka, Reconstructing graphs from smaller subgraphs
2019-10-10 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
A graph or graph property is $ell$-reconstructible if it is determined by the multiset of all subgraphs obtained by deleting $ell$ vertices. Apart from the famous Graph Reconstruction Conjecture, Kelly conjectured in 1957 that for each $ellinmathbb N$, there is an integer $n=n(ell)$ such that every graph with at least $n$ vertices is $ell$-reconstructible.
We show that for each $nge7$ and every $n$-vertex graph $G$, the degree list and connectedness of $G$ are $3$-reconstructible, and the threshold $ngeq 7$ is sharp for both properties.‌ We also show that all $3$-regular graphs are $2$-reconstructible.

Alexandr V. Kostochka, On Ramsey-type problems for paths and cycles in dense graphs
2019-10-08 16:30 - 17:30 | Room 1501, Bldg. E6-1, KAIST
A well-known Ramsey-type puzzle for children is to prove that among any 6 people either there are 3 who know each other or there are 3 who do not know each other. More generally, a graph $G$ arrows a graph $H$ if for any coloring of the edges of $G$ with two colors, there is a monochromatic copy of $H$. In these terms, the above puzzle claims that the complete $6$-vertex graph $K_6$ arrows the complete $3$-vertex graph $K_3$.
We consider sufficient conditions on the dense host graphs $G$ to arrow long paths and even cycles. In particular, for large $n$ we describe all multipartite graphs that arrow paths and cycles with $2n$ edges. This implies a conjecture by Gyárfás, Ruszinkó, Sárkőzy and Szemerédi from 2007 for such $n$. Also for large $n$ we find which minimum degree in a $(3n-1)$-vertex graph $G$ guarantees that $G$ arrows the $2n$-vertex path. This yields a more recent conjecture of Schelp.
This is joint work with Jozsef Balogh, Mikhail Lavrov and Xujun Liu.

Casey Tompkins, Extremal problems for Berge hypergraphs
2019-10-01 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
Given a graph $G$, there are several natural hypergraph families one can define. Among the least restrictive is the family $BG$ of so-called Berge copies of the graph $G$. In this talk, we discuss Turán problems for families $BG$ in $r$-uniform hypergraphs for various graphs $G$. In particular, we are interested in general results in two settings: the case when $r$ is large and $G$ is any graph where this Turán number is shown to be eventually subquadratic, as well as the case when $G$ is a tree where several exact results can be obtained. The results in the first part are joint with Grósz and Methuku, and the second part with Győri, Salia and Zamora.

Cory Palmer, A survey of Turán-type subgraph counting problems
2019-09-19 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
Let $F$ and $H$ be graphs. The subgraph counting function $operatorname{ex}(n,H,F)$ is defined as the maximum possible number of subgraphs $H$ in an $n$-vertex $F$-free graph. This function is a direct generalization of the Turán function as $operatorname{ex}(n,F)=operatorname{ex}(n,K_2,F)$. The systematic study of $operatorname{ex}(n,H,F)$ was initiated by Alon and Shikhelman in 2016 who generalized several classical results in extremal graph theory to the subgraph counting setting. Prior to their paper, a number of individual cases were investigated; a well-known example is the question to determine the maximum number of pentagons in a triangle-free graph. In this talk we will survey results on the function $operatorname{ex}(n,H,F)$ including a number of recent papers. We will also discuss this function’s connection to hypergraph Turán problems.

Kevin Hendrey, The minimum connectivity forcing forest minors in large graphs
2019-09-10 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
Given a graph $G$, we define $textrm{ex}_c(G)$ to be the minimum value of $t$ for which there exists a constant $N(t,G)$ such that every $t$-connected graph with at least $N(t,G)$ vertices contains $G$ as a minor. The value of $textrm{ex}_c(G)$ is known to be tied to the vertex cover number $tau(G)$, and in fact $tau(G)leq textrm{ex}_c(G)leq frac{31}{2}(tau(G)+1)$. We give the precise value of $textrm{ex}_c(G)$ when $G$ is a forest. In particular we find that $textrm{ex}_c(G)leq tau(G)+2$ in this setting, which is tight for infinitely many forests.

Mihyun Kang (강미현), The genus of a random graph and the fragile genus property
2019-08-20 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
In this talk we shall discuss how quickly the genus of the Erdős-Rényi random graph grows as the number of edges increases and how dramatically a small number of random edges can increase the genus of a randomly perturbed graph. (Joint work with Chris Dowden and Michael Krivelevich)

2019 Combinatorics Workshop
2019-08-13 - 2019-08-15 | Hotel Skypark, Songdo, Incheon, Korea (송도 스카이파크호텔)
The annual conference on Combinatorics Workshop (조합론 학술대회) began in 2004 by the Yonsei University BK21 Research Group. This year it will take place in Songdo, Incheon, August 13-15, 2019.

Due to the capacity (50 persons) of the place, we are able to limit your registration. In principle, registration is on a first-come, first-served basis.

Title 2019 Combinatorics Workshop (2019 조합론 학술대회)
Date August 13-15 (Tue-Thu), 2019
Venue Hotel Skypark, Songdo, Incheon
Web https://cw2019.combinatorics.kr

We are going to

give six 50-minute talks and ten 25-minute talks in Korean.
distribute the program and abstracts of CW2019.
provide for the accommodations for all participants.
provide for six meals (two breakfasts, two luncheons, one dinner, and one banquet) for all participants.

Plenary Speaker

Mihyun Kang, Graz University of Technology

Keynote Speakers

Jaehoon Kim, KAIST
Seung Jin Lee, Seoul National University
Meesue Yoo, Dankook University

Invited Speakers

Eun-Kyung Cho, Hankuk University of Foreign Studies
Soogang Eoh, Seoul National University
JiSun Huh, Sungkyunkwan University
Jihyeug Jang, Sungkyunkwan University
Dong Yeap Kang, KAIST and IBS Discrete Mathematics Group
Min Jeong Kang, Incheon National University
Dabeen Lee, IBS Discrete Mathematics Group
Kang-Ju Lee, Seoul National University
Myeonghwan Lee, Incheon National University
Jihye Park, Yeungnam University
Sun-mi Yun, Sungkyunkwan University

Organizing Committee

O-joung Kwon, Incheon National University and IBS Discrete Mathematics Group
Suil O, SUNY Korea
Sang-il Oum, IBS Discrete Mathematics Group and KAIST
Heesung Shin, Inha University

Advisory Committee

Committee of Discrete Mathematics, The Korean Mathematical Society (Chair: Seog-Jin Kim, Konkuk University)

Sponsors

IBS Discrete Mathematics Group
NRF (National Research Foundation of Korea)

2019-2 IBS One-Day Conference on Extremal Graph Theory
2019-08-12 11:00 - 20:00 | Room B232, IBS (기초과학연구원)
Invited Speakers

Jaehoon Kim (김재훈), KAIST
Hong Liu (刘鸿), University of Warwick
Abhishek Methuku, IBS Discrete Mathematics Group
Péter Pál Pach, Budapest University of Technology and Economics

Schedule
August 12, Monday
11:00am-12:00pm Jaehoon Kim (김재훈): Tree decompositions of graphs without large bipartite holes
12:00pm-1:30pm Lunch
1:30pm-2:30pm Abhishek Methuku: Bipartite Turán problems for ordered graphs
3:00pm-4:00pm Péter Pál Pach: On some applications of graph theory to number theoretic problems
4:30pm-5:30pm Hong Liu: Recent advance in Ramsey-Turán theory
6:00pm-8:00pm Banquet
Abstract
Jaehoon Kim (김재훈), Tree decompositions of graphs without large bipartite holes
A recent result of Condon, Kim, Kühn and Osthus implies that for any $rgeq (frac{1}{2}+o(1))n$, an $n$-vertex almost $r$-regular graph $G$ has an approximate decomposition into any collections of $n$-vertex bounded degree trees. In this talk, we prove that a similar result holds for an almost $alpha n$-regular graph $G$ with any $alpha>0$ and a collection of bounded degree trees on at most $(1-o(1))n$ vertices if $G$ does not contain large bipartite holes. This result is sharp in the sense that it is necessary to exclude large bipartite holes and we cannot hope for an approximate decomposition into $n$-vertex trees. This is joint work with Younjin Kim and Hong Liu.
Abhishek Methuku, Bipartite Turán problems for ordered graphs
A zero-one matrix $M$ contains a zero-one matrix $A$ if one can delete rows and columns of $M$, and turn 1-entries into 0-entries such that the resulting matrix is $A$. The extremal number of $A$, denoted by $ex(n,A)$, is the maximum number of $1$-entries in an $ntimes n$ sized matrix $M$ that does not contain $A$.
A matrix $A$ is column-$t$-partite (or row-$t$-partite), if it can be cut along the columns (or rows) into $t$ submatrices such that every row (or column) of these submatrices contains at most one $1$-entry. We prove that if $A$ is column-$t$-partite, then $operatorname{ex}(n,A)<n^{2-frac{1}{t}+frac{1}{2t^{2}}+o(1)}$, and if $A$ is both column- and row-$t$-partite, then $operatorname{ex}(n,A)<n^{2-frac{1}{t}+o(1)}$, and this bound is close to being optimal. Our proof introduces a new density-increment-type argument which is combined with the celebrated dependent random choice method.
Results about the extremal numbers of zero-one matrices translate into results about the Turán numbers of bipartite ordered graphs. In particular, a zero-one matrix with at most $t$ 1-entries in each row corresponds to an ordered graph with maximum degree $t$ in one of its vertex classes. The aim of this talk is to establish general results about the extremal numbers of ordered graphs. Our results are partially motivated by a well known result of Füredi (1991) and Alon, Krivelevich, Sudakov (2003) stating that if $H$ is a bipartite graph with maximum degree $t$ in one of the vertex classes, then $operatorname{ex}(n,H)=O(n^{2-frac{1}{t}})$. This is joint work with Tomon.
Péter Pál Pach, On some applications of graph theory to number theoretic problems
How large can a set of integers be, if the equation $a_1a_2dots a_h=b_1b_2dots b_h$ has no solution consisting of distinct elements of this set? How large can a set of integers be, if none of them divides the product of $h$ others? How small can a multiplicative basis for ${1, 2, dots, n}$ be? The first question is about a generalization of the multiplicative Sidon sets, the second one is of the primitive sets, while the third one is the multiplicative version of the well-studied analogue problem for additive bases.
In answering the above mentioned questions graph theory plays an important role and in most of our results not only the asymptotics are found, but very tight bounds are obtained for the error terms, as well. We will also discuss the counting version of these questions.
Hong Liu, Recent advance in Ramsey-Turán theory
We will talk about Ramsey-Turán theory and some recent development. More specifically, we will talk about graphs with $alpha_r(G)=o(n)$, where $alpha_r(G)$ is the largest size subset with no $K_r$. Two major open problems in this area from the 80s ask: (1) whether Bollobas-Erdos graph can be generalised to densities other than $1/2$; (2) whether the asymptotic extremal structure for $alpha_r(G)$ resembles that of $alpha_2(G)$. We settle the first conjecture by constructing Bollobas-Erdos type graph with density $a$ for arbitrary rational number $ale 1/2$; and refute the second conjecture by witnessing asymptotic extremal structures that are drastically different from the $alpha_2(G)$ problem via a ”Mobius product” construction.
Joint work with Christian Reiher, Maryam Sharifzadeh, and Katherine Staden.

2019 IBS Summer Research Program on Algorithms and Complexity in Discrete Structures
2019-07-21 - 2019-08-10 | Room B232, IBS (기초과학연구원)
Schedule
July 22 Monday
10:00-11:00 Introduction, 11:00-12:00 Open Problems
July 23 Tuesday
10:00-10:30 Stefan Kratsch, Humboldt-Universität zu Berlin, Germany
Elimination Distances, Blocking Sets, and Kernels for Vertex Cover
10:45-11:15 Benjamin Bergougnoux, University Clermont Auvergne, France
More applications of the d-neighbor equivalence
11:30-12:00 Yixin Cao, Hong Kong Polytechnic University, China
Enumerating Maximal Induced Subgraphs
13:30-14:30 Open Problems
July 24 Wednesday
We will go to KAIST for June Huh‘s two talks 15:00-16:00, 16:30-17:30. A IBS shuttle bus @14:30 from the lobby.
July 25 Thursday
10:00-11:00 Nick Brettell, Durham University, UK
Recent work on characterising matroids representable over finite fields
11:30-12:00 Mamadou M. Kanté, University Clermont Auvergne, France
On recognising k-letter graphs
July 26 Friday
10:00-11:00 O-joung Kwon (권오정), Incheon National University, Korea
The grid theorem for vertex-minors
11:00-12:00 Progress Report
July 27 Saturday
8:30 Excursion to Damyang (departure from the accommodation)
July 29 Monday
10:00-11:00 Archontia Giannopoulou, National and Kapodistrian University of Athens, Greece
The directed flat wall theorem
11:30-12:00 Eunjung Kim (김은정), LAMSADE-CNRS, France
Subcubic even-hole-free graphs have a constant treewidth
July 30 Tuesday
10:00-11:00 Pierre Aboulker, ENS Ulm, France
Generalizations of the geometric de Bruijn Erdős Theorem
11:30-12:00 Michael Dobbins, Binghamton University, USA
Barycenters of points in polytope skeleta
July 31 Wednesday
10:00-11:00 Magnus Wahlström, Royal Holloway, University of London, UK
FPT-algorithms via LP-relaxations
11:30-12:00 Édouard Bonnet, ENS Lyon, France
The FPT/W[1]-hard dichotomy of Max Independent Set in H-free graphs
August 1 Thursday
10:00-11:00 Euiwoong Lee (이의웅), NYU, USA
Losing treewidth by separating subsets
11:30-12:00 Sang-il Oum (엄상일), IBS Discrete Mathematics Group and KAIST, Korea
Branch-depth: Generalizing tree-depth of graphs
August 2 Friday
10:00-11:00 Dabeen Lee (이다빈), IBS Discrete Mathematics Group, Korea
t-perfect graphs and the stable set problem
11:00-12:00 Progress Report
August 5-9: Free Discussions / Research Collaborations / Progress Report
Tea time: Every weekday 3:30pm
Abstracts
July 23 Tuesday
Stefan Kratsch, Elimination Distances, Blocking Sets, and Kernels for Vertex Cover
The Vertex Cover problem plays an essential role in the study of polynomial kernelization in parameterized complexity, i.e., the study of provable and efficient preprocessing for NP-hard problems. Motivated by the great variety of positive and negative results for kernelization for Vertex Cover subject to different parameters and graph classes, we seek to unify and generalize them using so-called blocking sets, which have played implicit and explicit roles in many results. We show that in the most-studied setting, parameterized by the size of a deletion set to a specified graph class ${cal C}$, bounded minimal blocking set size is necessary but not sufficient to get a polynomial kernelization. Under mild technical assumptions, bounded minimal blocking set size is showed to allow an essentially tight efficient reduction in the number of connected components. We then determine the exact maximum size of minimal blocking sets for graphs of bounded elimination distance to any hereditary class ${cal C}$, including the case of graphs of bounded treedepth. We get similar but not tight bounds for certain non-hereditary classes ${cal C}$, including the class ${cal C}_{LP}$ of graphs where integral and fractional vertex cover size coincide. These bounds allow us to derive polynomial kernels for Vertex Cover parameterized by the size of a deletion set to graphs of bounded elimination distance to, e.g., forest, bipartite, or ${cal C}_{LP}$ graphs.
Benjamin Bergougnoux, More applications of the d-neighbor equivalence
In this talk, I will present a framework which gives efficient algorithms for several connectivity problems such as Connected Dominating Set, Node Weighted Steiner Tree, Maximum Induced Tree, Longest Induced Path, and Feedback Vertex Set. For all these problems, we obtain efficient algorithms parameterized respectively by tree-width, clique-width, Q-rank-width, rank-width and maximum induced matching width. Our approach simplifies and unifies the known algorithms for each of the parameters and match asymptotically also the running time of the best algorithms for Vertex Cover and Dominating Set. Our framework is based on the d-neighbor equivalence defined in [Bui-Xuan, Telle and Vatshelle, TCS 2013]. The results we obtain highlight the importance and the generalizing power of this equivalence relation on width measures. We also prove that this equivalence relation could be useful for Max Cut: a W[1]-hard problem parameterized by clique-width.
Paper available on HAL : https://hal.archives-ouvertes.fr/hal-01799573
Yixin Cao, Enumerating Maximal Induced Subgraphs
We study efficient algorithms for the maximal induced subgraphs problem: Given a graph on $n$ vertices, enumerate all its maximal induced subgraphs that belong to a particular hereditary graph class. While its optimization version, known as the vertex deletion problem in literature, has been intensively studied, algorithms for the enumeration version exist only for few simple graph classes, e.g., independent sets and cliques. Enumeration algorithms are mainly categorized in three complexity classes, polynomial total time (polynomial on $n$ and the total number of solutions), incremental polynomial time (for all $s$, the time to output the first $s$ solutions is polynomial on $n$ and $s$), and polynomial delay (for all $s$, the time to output the first $s$ solutions is polynomial on $n$ and linear on $s$). We first show that the maximal induced subgraphs problem can be solved in polynomial total time if and only if it can be solved in incremental polynomial time. We then propose general approaches to develop polynomial-delay algorithms and incremental-polynomial-time algorithms for the problem. They enable us to develop simple algorithms to solve the problem with polynomial delay for a large number of graph classes, and in incremental polynomial time for interval graphs, chordal graphs as well as two of its well-studied subclasses, and all graph classes with finite forbidden induced subgraphs.
July 25 Thursday
Nick Brettell, Recent work on characterising matroids representable over finite fields
Characterising when a matroid is representable over a certain field or set of fields is one of the oldest problems in matroid theory, dating back to Whitney’s 1935 paper. In this talk, I will discuss some of the more recent work in this area, with a focus on work I have been involved with towards characterising matroids representable over all fields of size at least four.
Mamadou M. Kanté, On recognising k-letter graphs
$k$-letter graphs are a subclass of graphs of linear clique-width $k$, but are well-quasi-ordered by the induced subgraph ordering. I present a simple FPT algorithm for recognising $k$-letter graphs and also an upper bound on the size of minimal obstructions. The characterisation is based on a simple observation allowing an MSOL-definability. (joint work with V. Lozin)
July 26 Friday
O-joung Kwon (권오정), The grid theorem for vertex-minors
We prove that, for a circle graph $H$, every graph with sufficiently large rank-width contains a vertex-minor isomorphic to $H$. This is joint work with Jim Geelen, Rose McCarty, and Paul Wollan.
July 29 Monday
Archontia Giannopoulou, The directed flat wall theorem
At the core of the Robertson-Seymour theory of Graph Minors lies a powerful structure theorem which captures, for any fixed graph $H$, the common structural features of all the graphs not containing $H$ as a minor [Neil Robertson, Paul D. Seymour: Graph Minors. XVI. Excluding a non-planar graph. J. Comb. Theory, Ser. B 89(1): 43-76 (2003)]. An important step towards this structure theorem is the Flat Wall Theorem [Neil Robertson, Paul D. Seymour: Graph Minors .XIII. The Disjoint Paths Problem. J. Comb. Theory, Ser. B 63(1): 65-110 (1995)], which has a lot of algorithmic applications (for example, the minor-testing and the disjoint paths problem with fixed number terminals).
In this paper, we prove the directed analogue of this Flat Wall Theorem. Our result builds on the recent Directed Grid Theorem by two of the authors (Kawarabayashi and Kreutzer), and we hope that this is an important and significant step toward the directed structure theorem, as with the case for the undirected graph for the graph minor project.
Joint work with Ken-ichi Kawarabayashi, Stephan Kreutzer, and O-joung Kwon.
Eunjung Kim (김은정), Subcubic even-hole-free graphs have a constant treewidth
It is known that even-hole-free graphs can have arbitrarily large rankwidth, but all known constructions have many high-degree vertices. It has been believed in the structural graph theory community that the rankwidth shall be bounded if the maximum degree is bounded. We prove that there exists a universal constant $c$ such that every even-hole-free graph of degree at most three has treewidth at most $c$. As a by-product of the proof, we also show that there exists a function $f$ such that for any $r$, $K_r$-minor-free and even-hole-free graph has treewidth at most $f(r)$; this extends the result of Silva et. al. (Discrete Applied Mathematics 2010) addressing the case of planar graphs.
This is a joint work with Pierre Aboulker and Isolde Adler.
July 30 Tuesday
Pierre Aboulker, Generalizations of the geometric de Bruijn Erdős Theorem
A classic Theorem of de Bruijn and Erdős states that every noncollinear set of n points in the plane determines at least n distinct lines. The line $L(u,v)$ determined by two points $u$, $v$ in the plane consists of all points $p$ such that

$operatorname{dist}(p, u)+operatorname{dist}(u, v)=operatorname{dist}(p, v)$ (i.e. $u$ is between $p$ and $v$) or
$operatorname{dist}(u, p)+operatorname{dist}(p, v)=operatorname{dist}(u, v)$ (i.e. $p$ is between $u$ and $v$) or
$operatorname{dist}(u, v)+operatorname{dist}(v, p)=operatorname{dist}(u, p)$ (i.e. $v$ is between $u$ and $p$).

With this definition of line $L(uv)$ in an arbitrary metric space $(V, operatorname{dist})$, Chen and Chvátal conjectured that every metric space on n points, where n is at least 2, has at least n distinct lines or a line that consists of all n points. The talk will survey results on and around this conjecture.
Michael Dobbins, Barycenters of points in polytope skeleta
In this talk I will classify the n-tuples of dimensions such that any target point in any d-polytope is the barycenter of points in faces of the prescribed dimensions. Specifically, for a $(nk+r)$-polytope and target point, we can always find $r$ points in faces of dimension $k+1$, and $n-r$ points in faces of dimension $k$, that have their barycenter at the given target. This is tight in the sense that we cannot require even one of the points to be in a face of dimension less than $k$, and we cannot require more than $n-r$ of the points to be in faces of dimension $k$. This is joint work with Florian Frick.
July 31 Wednesday
Magnus Wahlström, FPT-algorithms via LP-relaxations
LP-relaxations are traditionally (within theoretical computer science) used for computing approximate solutions to NP-hard problems, but across the last few years there have been several examples where LP-relaxations have been used to guide FPT-algorithms — that is, exact (non-approximate) algorithms whose running time is bounded in terms of a “parameter” of the input instance. This approach has given algorithms that are simultaneously simpler and faster for a range of central problems in parameterized complexity. At the same time, this is applicable only to specific problems and relaxations; an arbitrary LP-relaxation, even one that has good properties in an approximation sense, will in general give no useful guidance with respect to exact solutions.
In this talk, we give an overview of these FPT applications, and the conditions they impose on the LP-relaxation. Our main focus is on an approach of “discrete relaxation” referred to as Valued CSPs, but we also briefly survey more immediately combinatorial conditions, as well as related algorithms that solve these problems more efficiently (e.g., so-called linear-time FPT algorithms) by bypassing the LP-solver.
Édouard Bonnet, The FPT/W[1]-hard dichotomy of Max Independent Set in H-free graphs
Maximum Independent Set (MIS) is in general graphs the paradigmatic W[1]-hard problem. In stark contrast, polynomial-time algorithms are known when the inputs are restricted to structured graph classes such as, for instance, perfect graphs (which includes bipartite graphs, chordal graphs, co-graphs, etc.) or claw-free graphs. In this talk, we introduce some variants of co-graphs with “parameterized noise”, that is, graphs that can be turned into disjoint unions or complete sums by the removal of a certain number of vertices and the edition of a certain number of edges per incident vertex, both controlled by the parameter. We give a series of FPT Turing-reductions on these classes and use them to make some progress on the parameterized complexity of MIS in $H$-free graphs. We present the current state of the (randomized) FPT vs W[1]-hard dichotomy of MIS in $H$-free graphs. We then show that for every fixed $t geq 1$, MIS is FPT in $P(1,t,t,t)$-free graphs, where $P(1,t,t,t)$ is the graph obtained by substituting all the vertices of a four-vertex path but one end of the path by cliques of size t. In particular, this helped finishing the classification when H has at most five vertices.
This follows joint works with Nicolas Bousquet, Pierre Charbit, Stéphan Thomassé, and Rémi Watrigant.
August 1 Thursday
Euiwoong Lee (이의웅), Losing treewidth by separating subsets
We study the problem of deleting the smallest set S of vertices (resp. edges) from a given graph $G$ such that the induced subgraph (resp. subgraph) $G setminus S$ belongs to some class of graphs $H$. When graphs in H have treewidth bounded by $t$, we construct a framework for both vertex and edge deletion versions where approximation algorithms for these problems can be obtained from approximation algorithms for natural graph partitioning problems called $k$-Subset Vertex Separator and $k$-Subset Edge Separator.
For the vertex deletion, our framework, combined with the previous best result for $k$-Subset Vertex Separator, yields improved approximation ratios for fundamental problems such as $k$-Treewidth Vertex Deletion and Planar-$F$ Vertex Deletion. For the edge deletion, we give improved approximation algorithms for $k$-Subset Edge Separator and their applications to various problems in bounded-degree graphs.
Joint work with Anupam Gupta, Jason Li, Pasin Manurangsi, and Michal Wlodarczyk.
Sang-il Oum (엄상일), Branch-depth: Generalizing tree-depth of graphs
We present a concept called the branch-depth of a connectivity function, that generalizes the tree-depth of graphs. Then we prove two theorems showing that this concept aligns closely with the notions of tree-depth and shrub-depth of graphs as follows. For a graph $G = (V,E)$ and a subset $A $ of $E$ we let $lambda_G (A)$ be the number of vertices incident with an edge in $A$ and an edge in $E setminus A$. For a subset $X$ of $V$, let $rho_G(X)$ be the rank of the adjacency matrix between $X$ and $V setminus X$ over the binary field. We prove that a class of graphs has bounded tree-depth if and only if the corresponding class of functions $lambda_G$ has bounded branch-depth and similarly a class of graphs has bounded shrub-depth if and only if the corresponding class of functions $rho_G$ has bounded branch-depth, which we call the rank-depth of graphs.
Furthermore we investigate various potential generalizations of tree-depth to matroids and prove that matroids representable over a fixed finite field having no large circuits are well-quasi-ordered by the restriction.
This is a joint work with Matt DeVos and O-joung Kwon.
August 2 Friday
Dabeen Lee (이다빈), t-perfect graphs and the stable set problem
A graph $G$ is called $t$-perfect if the stable set polytope of $G$ can be obtained after adding the odd cycle inequalities to the standard edge formulation for the problem. A motivation for studying $t$-perfect graphs is algorithmic: as the odd cycle inequalities can be separated in polynomial time, one can compute a maximum weight stable set in a $t$-perfect graph by an LP algorithm in polynomial time. There are rich classes of $t$-perfect graphs, but a complete characterization of $t$-perfect graphs is still open. Due to this, no graph-theoretic / combinatorial algorithm is known, although there is an algorithm for the unweighted case based on a combinatorial approximate LP algorithm. In this survey talk, I will (i) review classes of known $t$-perfect graphs, (ii) briefly explain the algorithm for the unweighted case and a possibility of extending it to the weighted case, and (iii) discuss some research directions towards finding a graph-theoretic algorithm for computing a maximum weight stable set in a $t$-perfect graph.
Participants (not including IBS researchers)

Pierre Aboulker, ENS Ulm, France
Rémy Belmonte, University of Electro-Communications, Japan
Benjamin Bergougnoux, University of Paris Diderot, France
Édouard Bonnet, ENS Lyon, France
Nick Brettell, Durham University, UK
Yixin Cao, Hong Kong Polytechnic University, China
Michael Dobbins, Binghamton University, USA
Chris Eppolito, Binghamton University, USA
Archontia Giannopoulou, National and Kapodistrian University of Athens, Greece
Mamadou M. Kante, University Clermont Auvergne, France
Eunjung Kim (김은정), LAMSADE-CNRS, France
Stefan Kratsch, Humboldt-Universität zu Berlin, Germany
O-joung Kwon (권오정), Incheon National University, Korea
Michael Lampis, Université Paris-Dauphine, France
Euiwoong Lee (이의웅), NYU, USA
Valia Mitsou, Université Paris-Diderot, France
Florian Sikora, University Paris Dauphine, France
Magnus Wahlström, Royal Holloway, University of London, UK

An invitation-only summer research program will be held in the summer of 2019. There will be 10-20 participants to work together at DIMAG. Talks are open.
Accommodation
Everyone will stay at the Daedeok Innopolis Guest House (대덕특구게스트하우스).

Address: 27-5, Expo-ro 123beon-gil, Yuseong-gu, Daejeon, 34125, Republic of Korea (대전광역시 유성구 엑스포로 123번길 27-5)
Tel: (+82)-042-865-2500

Organizers

Eunjung Kim (김은정), LAMSADE-CNRS, France
Sang-il Oum (엄상일), IBS Discrete Mathematics Group and KAIST, Korea

Dabeen Lee (이다빈), Integrality of set covering polyhedra and clutter minors
2019-07-16 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
Given a finite set of elements $V$ and a family $mathcal{C}$ of subsets of $V$, the set covering problem is to find a minimum cardinality subset of $V$ intersecting every subset in the family $mathcal{C}$. The set covering problem, also known as the hitting set problem, admits a simple integer linear programming formulation. The constraint system of the integer linear programming formulation defines a polyhedron, and we call it the set covering polyhedron of $mathcal{C}$. We say that a set covering polyhedron is integral if every extreme point is an integer lattice point. Although the set covering problem is NP-hard in general, conditions under which the problem becomes polynomially solvable have been studied. If the set covering polyhedron is integral, then it is straightforward that the problem can be solved using a polynomial-time algorithm for linear programming.
In this talk, we will focus on the question of when the set covering polyhedron is integral. We say that the family $mathcal{C}$ is a clutter if every subset in $mathcal{C}$ is inclusion-wise minimal. As taking out non-minimal subsets preserves integrality, we may assume that $mathcal{C}$ is a clutter. We call $mathcal{C}$ ideal if the set covering polyhedron of it is integral. To understand when a clutter is ideal, the notion of clutter minors is important in that $mathcal{C}$ is ideal if and only if so is every minor of it. We will study two fundamental classes of non-ideal clutters, namely, deltas and the blockers of extended odd holes. We will characterize when a clutter contains either a delta or the blocker of an extended odd hole as a minor.
This talk is based on joint works with Ahmad Abdi and Gérard Cornuéjols.

Patrice Ossona de Mendez, A model theoretical approach to sparsity
2019-06-25 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
We discuss how the model theoretic notion of first-order transduction allows to define a notion of structural sparsity, and give some example of applications, like existence of low shrub-depth decompositions for tranductions of bounded expansion classes, characterization of transductions of classes with bounded pathwidth, decompositions of graphs with bounded rank-width into cographs.

Suil O (오수일), An odd [1,b]-factor in regular graphs from eigenvalues
2019-06-19 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
An odd $[1,b]$-factor of a graph is a spanning subgraph $H$ such that for every vertex $v in V(G)$, $1 le d_H(v) le b$, and $d_H(v)$ is odd. For positive integers $r ge 3$ and $b le r$, Lu, Wu, and Yang gave an upper bound for the third largest eigenvalue in an $r$-regular graph with even number of vertices to guarantee the existence of an odd [1,b]-factor.
In this talk, we improve their bound.

Jinyoung Park (박진영), The number of maximal independent sets in the Hamming cube
2019-06-03 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
Let $Q_n$ be the $n$-dimensional Hamming cube (hypercube) and $N=2^n$. We prove that the number of maximal independent sets in $Q_n$ is asymptotically $2n2^{N/4}$, as was conjectured by Ilinca and Kahn in connection with a question of Duffus, Frankl and Rödl. The value is a natural lower bound derived from a connection between maximal independent sets and induced matchings. The proof of the upper bound draws on various tools, among them “stability” results for maximal independent set counts and old and new results on isoperimetric behavior in $Q_n$. This is joint work with Jeff Kahn.

Lars Jaffke, A complexity dichotomy for critical values of the b-chromatic number of graphs
2019-05-20 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
A $b$-coloring of a graph $G$ is a proper coloring of its vertices such that each color class contains a vertex that has at least one neighbor in all the other color classes. The $b$-Coloring problem asks whether a graph $G$ has a $b$-coloring with $k$ colors.
The $b$-chromatic number of a graph $G$, denoted by $chi_b(G)$, is the maximum number $k$ such that $G$ admits a $b$-coloring with $k$ colors. We consider the complexity of the $b$-Coloring problem, whenever the value of $k$ is close to one of two upper bounds on $chi_b(G)$: The maximum degree $Delta(G)$ plus one, and the $m$-degree, denoted by $m(G)$, which is defined as the maximum number $i$ such that $G$ has $i$ vertices of degree at least $i-1$. We obtain a dichotomy result stating that for fixed $k in{Delta(G) + 1 − p, m(G) − p}$, the problem is polynomial-time solvable whenever $pin{0, 1}$ and, even when $k = 3$, it is NP-complete whenever $p ge 2$.
We furthermore consider parameterizations of the $b$-Coloring problem that involve the maximum degree $Delta(G)$ of the input graph $G$ and give two FPT-algorithms. First, we show that deciding whether a graph G has a $b$-coloring with $m(G)$ colors is FPT parameterized by $Delta(G)$. Second, we show that $b$-Coloring is FPT parameterized by $Delta(G) + ell_k(G)$, where $ell_k(G)$ denotes the number of vertices of degree at least $k$.
This is joint work with Paloma T. Lima.

Xin Zhang (张欣), On equitable tree-colorings of graphs
2019-05-16 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
An equitable tree-$k$-coloring of a graph is a vertex coloring using $k$ distinct colors such that every color class (i.e, the set of vertices in a common color) induces a forest and the sizes of any two color classes differ by at most one. The minimum integer $k$ such that a graph $G$ is equitably tree-$k$-colorable is the equitable vertex arboricity of $G$, denoted by $va_{eq}(G)$. A graph that is equitably tree-$k$-colorable may admits no equitable tree-$k’$-coloring for some $k’>k$. For example, the complete bipartite graph $K_{9,9}$ has an equitable tree-$2$-coloring but is not equitably tree-3-colorable. In view of this a new chromatic parameter so-called the equitable vertex arborable threshold is introduced. Precisely, it is the minimum integer $k$ such that $G$ has an equitable tree-$k’$-coloring for any integer $k’geq k$, and is denoted by $va_{eq}^*(G)$. The concepts of the equitable vertex arboricity and the equitable vertex arborable threshold were introduced by J.-L. Wu, X. Zhang and H. Li in 2013. In 2016, X. Zhang also introduced the list analogue of the equitable tree-$k$-coloring. There are many interesting conjectures on the equitable (list) tree-colorings, one of which, for example, conjectures that every graph with maximum degree at most $Delta$ is equitably tree-$k$-colorable for any integer $kgeq (Delta+1)/2$, i.e, $va_{eq}^*(G)leq lceil(Delta+1)/2rceil$. In this talk, I review the recent progresses on the studies of the equitable tree-colorings from theoretical results to practical algorithms, and also share some interesting problems for further research.

Sang June Lee (이상준), On strong Sidon sets of integers
2019-05-08 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
Let $mathbb N$ be the set of natural numbers. A set $Asubset mathbb N$ is called a Sidon set if the sums $a_1+a_2$, with $a_1,a_2in S$ and $a_1leq a_2$, are distinct, or equivalently, if [|(x+w)-(y+z)|geq 1] for every $x,y,z,win S$ with $x<yleq z<w$. We define strong Sidon sets as follows:
For a constant $alpha$ with $0leq alpha<1$, a set $Ssubset mathbb N$ is called an $alpha$-strong Sidon set if [|(x+w)-(y+z)|geq w^alpha] for every $x,y,z,win S$ with $x<yleq z<w$.
The motivation of strong Sidon sets is that a strong Sidon set generates many Sidon sets by altering each element a bit. This infers that a dense strong Sidon set will guarantee a dense Sidon set contained in a sparse random subset of $mathbb N$.
In this talk, we are interested in how dense a strong Sidon set can be. This is joint work with Yoshiharu Kohayakawa, Carlos Gustavo Moreira and Vojtěch Rödl.

Rose McCarty, Circle graphs are polynomially chi-bounded
2019-04-26 16:00 - 17:00 | Room B232, IBS (기초과학연구원)
Circle graphs are the intersection graphs of chords on a circle; vertices correspond to chords, and two vertices are adjacent if their chords intersect. We prove that every circle graph with clique number k has chromatic number at most $4k^2$. Joint with James Davies.

Jon-Lark Kim (김종락), Introduction to Boolean functions with Artificial Neural Network
2019-04-18 11:00 - 12:00 | Room B232, IBS (기초과학연구원)
A Boolean function is a function from the set Q of binary vectors of length n (i.e., the binary n-dimensional hypercube) to $F_2={0,1}$. It has several applications to complexity theory, digital circuits, coding theory, and cryptography.
In this talk we give a connection between Boolean functions and Artificial Neural Network. We describe how to represent Boolean functions by Artificial Neural Network including linear and polynomial threshold units and sigmoid units. For example, even though a linear threshold function cannot realize XOR, a polynomial threshold function can do it. We also give currently open problems related to the number of (Boolean) linear threshold functions and polynomial threshold functions.

Andreas Holmsen, Large cliques in hypergraphs with forbidden substructures
2019-03-12 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
A result due to Gyárfás, Hubenko, and Solymosi, answering a question of Erdős, asserts that if a graph $G$ does not contain $K_{2,2}$ as an induced subgraph yet has at least $cbinom{n}{2}$ edges, then $G$ has a complete subgraph on at least $frac{c^2}{10}n$ vertices. In this paper we suggest a “higher-dimensional” analogue of the notion of an induced $K_{2,2}$, which allows us to extend their result to $k$-uniform hypergraphs. Our result also has interesting consequences in topological combinatorics and abstract convexity, where it can be used to answer questions by Bukh, Kalai, and several others.

2019-1 IBS Workshop on Graph Theory
2019-02-11 - 2019-02-12 | Room B234, IBS (기초과학연구원)
Invited Speakers

Jeong Han Kim (김정한), KIAS, Seoul
Martin Balko, Charles University, Prague
Dániel Gerbner, Alfréd Rényi Institute of Mathematics, Budapest
Cory T. Palmer, University of Montana, Missoula
Boram Park (박보람), Ajou University
Dong Yeap Kang (강동엽), KAIST

Schedule
Feb. 11, 2019, Monday
1:30pm-2:20pm Jeong Han Kim: Entropy and sorting
2:20pm-3:10pm Cory T. Palmer: Generalized Turán problems – Berge hypergraphs
Coffee Break
4:00pm-4:50pm Martin Balko: Ramsey numbers of edge-ordered graphs
4:50pm-5:40pm Dong Yeap Kang: On the rational Turán exponents conjecture
Banquet
Feb. 12, 2019, Tuesday
9:30am-10:20am Boram Park: Sum-free set problem on integers
Coffee Break
11:00am-11:50am Dániel Gerbner: Generalized Turán problems – counting subgraphs
Lunch
We plan to provide meals to all participants and provide a room at a near-by hotel for invited speakers and selected participants. Please register in the following form below by January 28, Monday; please register early if you want to receive the support for the accommodation.
Abstracts
Jeong Han Kim (김정한), Entropy and Sorting
We reconsider the old problem of sorting under partial information, and give polynomial time algorithms for the following tasks: (1) Given a partial order P, find (adaptively) a sequence of comparisons (questions of the form, “is x < y?”) which sorts ( i.e., finds an unknown linear extension of) P using O(log(e(P))) comparisons in worst case (where e(P) is the number of linear extensions of P). (2) Compute (on line) answers to any comparison algorithm for sorting a partial order P which force the algorithm to use Ω(log(e(P))) comparisons. (3) Given a partial order P of size n, estimate e(P) to within a factor exponential in n. (We give upper and lower bounds which differ by the factor $n^n /n!$.) Our approach, based on entropy of the comparability graph of P and convex minimization via the ellipsoid method, is completely different from earlier attempts to deal with these questions.
Joint work with J. Kahn.
Cory T. Palmer, Generalized Turán problems – Berge hypergraphs
Let $F$ be a graph. We say that a hypergraph $H$ is a Berge-$F$ if there is a bijection $f : E(F) rightarrow E(H )$ such that $e subseteq f(e)$ for every $e in E(F)$. Note that Berge-$F$ actually denotes a class of hypergraphs. The maximum number of edges in an $n$-vertex $r$-graph with no subhypergraph isomorphic to any Berge-$F$ is denoted $operatorname{ex}_r(n,textrm{Berge-}F)$. Observe that when $r=2$, then a Berge-$F$ is simply the graph $F$ and thus again we! are investigating the Tur’an function $operatorname{ex}(n,F)$.
In this talk we will survey results on the function $operatorname{ex}_r(n,textrm{Berge-}F)$ for various graphs $F$. We will also describe several interesting open problems.
Martin Balko, Ramsey numbers of edge-ordered graphs
An edge-ordered graph is a graph with linearly ordered set of edges. We introduce and study Ramsey numbers of edge-ordered graphs, called edge-ordered Ramsey numbers. We prove some basic properties of these numbers for general edge! -ordered graphs and we provide some stronger estimates for special classes of edge-ordered graphs. We also pose some new open problems and compare edge-ordered Ramsey numbers with the standard Ramsey numbers of graphs and with ordered Ramsey numbers, which are Ramsey numbers for graphs with linearly ordered vertex sets.
Joint work with Mate Vizer.
Dong Yeap Kang (강동엽), On the rational Turán exponents conjecture
The extremal number ${rm ex}(n,F)$ of a graph $F$ is the maximum number of edges in an $n$-vertex graph not containing $F$ as a subgraph. A real number $r in [1,2]$ is realisable if there exists a graph $F$ with ${rm ex}(n , F) = Theta(n^r)$. Several decades ago, Erdős and Simonovits conjectured that every rational number in $[1,2]$ is realisable. Despite decades of effort, the only known realisable numbers are $1,frac{7}{5},2$, and the numbers of the form $1+frac{1}{m}$, $2-frac{1}{m}$, $2-frac{2}{m}$ for integers $m geq 1$. In particular, it is not even known whether the set of all realisable numbers contains a single limit point other than two numbers 1 and 2.
We discuss some recent progress on the conjecture of Erdős and Simonovits. First, we show that $2 – frac{a}{b}$ is realisable for any integers $a,b geq 1$ with $b>a$ and $b equiv pm 1 ~({rm mod}:a)$. This includes all previously known ones, and gives infinitely many limit points $2-frac{1}{m}$ in the set of all realisable numbers as a consequence.
Secondly, we propose a conjecture on subdivisions of bipartite graphs. Apart from being interesting on its own, we show that, somewhat surprisingly, this subdivision conjecture in fact implies that every rational number between 1 and 2 is realisable.
This is joint work with Jaehoon Kim and Hong Liu.
Boram Park (박보람), Sum-free set problem on integers
For an abelian group $G$, a set $A subset G$ is sum-free if there are no $x, y, z in A$ such that $x + y = z$. Sum-free sets was initiated by Schur (1916) by an attempt to prove the famous Fermat’s Last Theorem. Since then, there have been intensive and fruitful research in the field of additive combinatorics. One of great interest in the study of sum-free sets is to consider sum-free subsets of a set of integers, which has attracted a significant attention in the literature over the years.
In this talk, some recent results on sum-free sets of integers are discussed. Then we present a result on $k$-sum $bf{n}$-free set, where $bf{n}$ is an $n$-dimensional integer vector. The work is based on joint work with Ilkyoo Choi and Ringi Kim.
Dániel Gerbner, Generalized Turán problems – counting subgraphs
Given two graphs $H$ and $F$, our goal is to determine the maximum number of copies of $H$ in an $F$-free graph on $n$ vertices. The systematic research of these problems was initiated (after several sporadic results) by Alon and Shikhelman. I describe several results of mine in this area, with different sets of co-authors.
Joint work with Ervin Győri, Abhishek Methuku, Cory Palmer and Mate Vizer.

Seog-Jin Kim (김석진), Signed colouring and list colouring of k-chromatic graphs
2019-01-28 16:00 - 17:00 | Room B232, IBS (기초과학연구원)
A signed graph is a pair (G, σ), where G is a graph and σ: E(G) → {1,-1} is a signature of G. A set S of integers is symmetric if I∈S implies that -i∈S. A k-colouring of (G,σ) is a mapping f:V(G) → Nk such that for each edge e=uv, f(x)≠σ(e) f(y), where Nk is a symmetric integer set of size k. We define the signed chromatic number of a graph G to be the minimum integer k such that for any signature σ of G, (G, σ) has a k-colouring.
Let f(n,k) be the maximum signed chromatic number of an n-vertex k-chromatic graph. This paper determines the value of f(n,k) for all positive integers n ≥ k. Then we study list colouring of signed graphs. A list assignment L of G is called symmetric if L(v) is a symmetric integer set for each vertex v. The weak signed choice number ch±w(G) of a graph G is defined to be the minimum integer k such that for any symmetric k-list assignment L of G, for any signature σ on G, there is a proper L-colouring of (G, σ). We prove that the difference ch±w(G)-χ±(G) can be arbitrarily large. On the other hand, ch±w(G) is bounded from above by twice the list vertex arboricity of G. Using this result, we prove that ch±w(K2⋆n)= χ±(K2⋆n) = ⌈2n/3⌉ + ⌊2n/3⌋. This is joint work with Ringi Kim and Xuding Zhu.

Eun Jung Kim (김은정), New algorithm for multiway cut guided by strong min-max duality
2019-01-04 16:00 - 17:00 | Room B232, IBS (기초과학연구원)
Problems such as Vertex Cover and Multiway Cut have been well-studied in parameterized complexity. Cygan et al. 2011 drastically improved the running time of several problems including Multiway Cut and Almost 2SAT by employing LP-guided branching and aiming for FPT algorithms parameterized above LP lower bounds. Since then, LP-guided branching has been studied in depth and established as a powerful technique for parameterized algorithms design.
In this talk, we make a brief overview of LP-guided branching technique and introduce the latest results whose parameterization is above even stronger lower bounds, namely μ(I)=2LP(I)-IP(dual-I). Here, LP(I) is the value of an optimal fractional solution and IP(dual-I) is the value of an optimal integral dual solution. Tutte-Berge formula for Maximum Matching (or equivalently Edmonds-Gallai decomposition) and its generalization Mader’s min-max formula are exploited to this end. As a result, we obtain an algorithm running in time 4k-μ(I) for multiway cut and its generalizations, where k is the budget for a solution.
This talk is based on a joint work with Yoichi Iwata and Yuichi Yoshida from NII.

Joonkyung Lee (이준경), Sidorenko’s conjecture for blow-ups
2019-01-03 16:00 - 17:00 | Room B232, IBS (기초과학연구원)
A celebrated conjecture of Sidorenko and Erdős–Simonovits states that, for all bipartite graphs H, quasirandom graphs contain asymptotically the minimum number of copies of H taken over all graphs with the same order and edge density. This conjecture has attracted considerable interest over the last decade and is now known to hold for a broad range of bipartite graphs, with the overall trend saying that a graph satisfies the conjecture if it can be built from simple building blocks such as trees in a certain recursive fashion.
Our contribution here, which goes beyond this paradigm, is to show that the conjecture holds for any bipartite graph H with bipartition A∪B where the number of vertices in B of degree k satisfies a certain divisibility condition for each k. As a corollary, we have that for every bipartite graph H with bipartition A∪B, there is a positive integer p such that the blow-up HAp formed by taking p vertex-disjoint copies of H and gluing all copies of A along corresponding vertices satisfies the conjecture. Joint work with David Conlon.

Hong Liu, Polynomial Schur’s Theorem
2018-12-13 17:00 - 18:00 | Room B109, IBS (기초과학연구원)
I will discuss the Ramsey problem for {x,y,z:x+y=p(z)} for polynomials p over ℤ. This is joint work with Peter Pach and Csaba Sandor.

Tony Huynh, A tight Erdős-Pósa function for planar minors
2018-12-10 17:00 - 18:00 | Room B109, IBS (기초과학연구원)
Let H be a planar graph. By a classical result of Robertson and Seymour, there is a function f(k) such that for all k and all graphs G, either G contains k vertex-disjoint subgraphs each containing H as a minor, or there is a subset X of at most f(k) vertices such that G−X has no H-minor. We prove that this remains true with f(k)=ck log k for some constant c depending on H. This bound is best possible, up to the value of c, and improves upon a recent bound of Chekuri and Chuzhoy. The proof is constructive and yields the first polynomial-time O(log 𝖮𝖯𝖳)-approximation algorithm for packing subgraphs containing an H-minor.
This is joint work with Wouter Cames van Batenburg, Gwenaël Joret, and Jean-Florent Raymond.
