Tony Huynh, A tight Erdős-Pósa function for planar minors

December 4th, 2018

IBS/KAIST Joint 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 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.

Ilkyoo Choi (최일규), Largest 2-regular subgraphs in 3-regular graphs

October 31st, 2018
Largest 2-regular subgraphs in 3-regular graphs
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, 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.

Jaehoon Kim, Introduction to Graph Decomposition

October 2nd, 2018
Introduction to Graph Decomposition
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.

Jaehoon Kim, Rainbow subgraphs in graphs

October 2nd, 2018
Rainbow subgraphs in graphs
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 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.

Dong Yeap Kang (강동엽), On the rational Turán exponents conjecture

September 27th, 2018
On the rational Turán exponents conjecture
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) = Θ(nr). 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.