Large cliques in hypergraphs with forbidden substructures

Andreas Holmsen

Department of Mathematical Sciences, KAIST

2019/03/12 Tue 4:30PM-5:30PM (Room B232, IBS)

A result due to Gyárfás, Hubenko, and Solymosi, answering a question of Erdős, asserts that if a graph G does not contain K2,2 as an induced subgraph yet has at least c n(n-1)/2 edges, then [...]

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]]>Large cliques in hypergraphs with forbidden substructures

Andreas Holmsen

Department of Mathematical Sciences, KAIST

Department of Mathematical Sciences, KAIST

2019/03/12 Tue 4:30PM-5:30PM (Room B232, IBS)

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 c n(n-1)/2 edges, then G has a complete subgraph on at least c^2 n /10 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.

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]]>IBS/KAIST Joint Discrete Math Seminar

Signed colouring and list colouring of k-chromatic graphs

Seog-Jin Kim (김석진)Department of Mathematics Education, Konkuk University, Seoul

2019/1/28 Mon 4PM-5PM (Room B232, IBS)

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 [...]

The post Seog-Jin Kim (김석진), Signed coloring and list coloring of k-chromatic graphs appeared first on KAIST Discrete Math Seminar.

]]>Signed colouring and list colouring of k-chromatic graphs

Seog-Jin Kim (김석진)

Department of Mathematics Education, Konkuk University, Seoul

Department of Mathematics Education, Konkuk University, Seoul

2019/1/28 Mon 4PM-5PM (Room B232, IBS)

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) → N_{k} such that for each edge e=uv, f(x)≠σ(e) f(y), where N_{k} 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}(K_{2⋆n})= χ_{±}(K_{2⋆n}) = ⌈2n/3⌉ + ⌊2n/3⌋. This is joint work with Ringi Kim and Xuding Zhu.

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]]>Sidorenko’s conjecture for blow-ups

Joonkyung Lee (이준경)

Universität Hamburg, Hamburg, Germany

2019/1/3 Thursday 4PM (Room: DIMAG, IBS)

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 [...]

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]]>Sidorenko’s conjecture for blow-ups

Joonkyung Lee (이준경)

Universität Hamburg, Hamburg, Germany

Universität Hamburg, Hamburg, Germany

2019/1/3 Thursday 4PM (Room: DIMAG, IBS)

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 H_{A}^{p} 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.

The post Joonkyung Lee (이준경), Sidorenko’s conjecture for blow-ups appeared first on KAIST Discrete Math Seminar.

]]>New algorithm for multiway cut guided by strong min-max duality

Eun Jung Kim (김은정)

CNRS, LAMSADE, Paris, France

2019/01/04 Fri 4PM-5PM (Room: DIMAG, IBS)

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 [...]

The post Eun Jung Kim (김은정), New algorithm for multiway cut guided by strong min-max duality appeared first on KAIST Discrete Math Seminar.

]]>New algorithm for multiway cut guided by strong min-max duality

Eun Jung Kim (김은정)

CNRS, LAMSADE, Paris, France

CNRS, LAMSADE, Paris, France

2019/01/04 Fri 4PM-5PM (Room: DIMAG, IBS)

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 4^{k-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.

The post Eun Jung Kim (김은정), New algorithm for multiway cut guided by strong min-max duality appeared first on KAIST Discrete Math Seminar.

]]>Polynomial Schur’s Theorem

Hong Liu

University of Warwick, UK

2018/12/13 Thu 5PM-6PM (Room B109, IBS)

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.

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]]>Polynomial Schur’s Theorem

Hong Liu

University of Warwick, UK

University of Warwick, UK

2018/12/13 Thu 5PM-6PM (Room B109, IBS)

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.

The post Hong Liu, Polynomial Schur’s Theorem appeared first on KAIST Discrete Math Seminar.

]]>A tight Erdős-Pósa function for planar minors

Tony Huynh

Université libre de Bruxelles

2018/12/10 5PM-6PM (Room B109, IBS)

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 [...]

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]]>A tight Erdős-Pósa function for planar minors

Tony Huynh

Université libre de Bruxelles

Université libre de Bruxelles

2018/12/10 5PM-6PM (Room B109, IBS)

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 ???)-approximation algorithm for packing subgraphs containing an H-minor.

This is joint work with Wouter Cames van Batenburg, Gwenaël Joret, and Jean-Florent Raymond.

The post Tony Huynh, A tight Erdős-Pósa function for planar minors appeared first on KAIST Discrete Math Seminar.

]]>Ilkyoo Choi (최일규)

Department of Mathematics, Hankuk University of Foreign Studies, Yongin-si

2018/11/26 Mon 5PM-6PM (Bldg. E6-1, Room 1401)

For a graph G, let f2(G) denote the largest number of vertices in a 2-regular subgraph of G. We determine the minimum of f2(G) over 3-regular n-vertex simple graphs G.

To do this, [...]

The post Ilkyoo Choi (최일규), Largest 2-regular subgraphs in 3-regular graphs appeared first on KAIST Discrete Math Seminar.

]]>Ilkyoo Choi (최일규)

Department of Mathematics, Hankuk University of Foreign Studies, Yongin-si

Department of Mathematics, Hankuk University of Foreign Studies, Yongin-si

2018/11/26 Mon 5PM-6PM (Bldg. E6-1, Room 1401)

For a graph G, let f_{2}(G) denote the largest number of vertices in a 2-regular subgraph of G. We determine the minimum of f_{2}(G) over 3-regular n-vertex simple graphs G.

To do this, we prove that every 3-regular multigraph with exactly c cut-edges has a 2-regular subgraph that omits at most max{0,⎣(c-1)/2⎦} vertices.

More generally, every n-vertex multigraph with maximum degree 3 and m edges has a 2-regular subgraph that omits at most max{0, ⎣(3n-2m+c-1)/2⎦} vertices.

These bounds are sharp; we describe the extremal multigraphs.

This is joint work with Ringi Kim, Alexandr V. Kostochka, Boram Park, and Douglas B. West.

To do this, we prove that every 3-regular multigraph with exactly c cut-edges has a 2-regular subgraph that omits at most max{0,⎣(c-1)/2⎦} vertices.

More generally, every n-vertex multigraph with maximum degree 3 and m edges has a 2-regular subgraph that omits at most max{0, ⎣(3n-2m+c-1)/2⎦} vertices.

These bounds are sharp; we describe the extremal multigraphs.

This is joint work with Ringi Kim, Alexandr V. Kostochka, Boram Park, and Douglas B. West.

The post Ilkyoo Choi (최일규), Largest 2-regular subgraphs in 3-regular graphs appeared first on KAIST Discrete Math Seminar.

]]>Jaehoon Kim (김재훈)

Mathematics Institute, University of Warwick, UK

2018/10/15 5PM

Graphs are mathematical structures used to model pairwise relations between objects.

Graph decomposition problems ask to partition the edges of large/dense graphs into small/sparse graphs.

In this talk, we introduce several famous graph decomposition problems, related puzzles and known results on the problems.

The post Jaehoon Kim, Introduction to Graph Decomposition appeared first on KAIST Discrete Math Seminar.

]]>Jaehoon Kim (김재훈)

Mathematics Institute, University of Warwick, UK

Mathematics Institute, University of Warwick, UK

2018/10/15 5PM

Graphs are mathematical structures used to model pairwise relations between objects.

Graph decomposition problems ask to partition the edges of large/dense graphs into small/sparse graphs.

In this talk, we introduce several famous graph decomposition problems, related puzzles and known results on the problems.

Graph decomposition problems ask to partition the edges of large/dense graphs into small/sparse graphs.

In this talk, we introduce several famous graph decomposition problems, related puzzles and known results on the problems.

The post Jaehoon Kim, Introduction to Graph Decomposition appeared first on KAIST Discrete Math Seminar.

]]>Jaehoon Kim (김재훈)

Mathematics Institute, University of Warwick, UK

2018/10/15 2:30PM

We say a subgraph H of an edge-colored graph is rainbow if all edges in H has distinct colors. The concept of rainbow subgraphs generalizes the concept of transversals in latin squares.

In this talk, we discuss how these concepts are related and we [...]

The post Jaehoon Kim, Rainbow subgraphs in graphs appeared first on KAIST Discrete Math Seminar.

]]>Jaehoon Kim (김재훈)

Mathematics Institute, University of Warwick, UK

Mathematics Institute, University of Warwick, UK

2018/10/15 2:30PM

We say a subgraph H of an edge-colored graph is rainbow if all edges in H has distinct colors. The concept of rainbow subgraphs generalizes the concept of transversals in latin squares.

In this talk, we discuss how these concepts are related and we introduce a result regarding approximate decompositions of graphs into rainbow subgraphs. This has implications on transversals in latin square. It is based on a joint work with Kühn, Kupavskii and Osthus.

In this talk, we discuss how these concepts are related and we introduce a result regarding approximate decompositions of graphs into rainbow subgraphs. This has implications on transversals in latin square. It is based on a joint work with Kühn, Kupavskii and Osthus.

The post Jaehoon Kim, Rainbow subgraphs in graphs appeared first on KAIST Discrete Math Seminar.

]]>Dong Yeap Kang (강동엽)

Department of Mathematical Sciences, KAIST

2018/11/5 Mon 5PM-6PM

The extremal number 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∈[1,2] is realisable if there exists a graph F with ex(n , F) = [...]

The post Dong Yeap Kang (강동엽), On the rational Turán exponents conjecture appeared first on KAIST Discrete Math Seminar.

]]>Dong Yeap Kang (강동엽)

Department of Mathematical Sciences, KAIST

Department of Mathematical Sciences, KAIST

2018/11/5 Mon 5PM-6PM

The extremal number 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∈[1,2] is *realisable* if there exists a graph F with ex(n , F) = Θ(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 1,7/5,2, and the numbers of the form 1+(1/m), 2-(1/m), 2-(2/m) for integers m≥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-(a/b) is realisable for any integers a,b≥1 with b>a and b≡±1 (mod a). This includes all previously known ones, and gives infinitely many limit points 2-(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.

We discuss some recent progress on the conjecture of Erdős and Simonovits. First, we show that 2-(a/b) is realisable for any integers a,b≥1 with b>a and b≡±1 (mod a). This includes all previously known ones, and gives infinitely many limit points 2-(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.

The post Dong Yeap Kang (강동엽), On the rational Turán exponents conjecture appeared first on KAIST Discrete Math Seminar.

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