Minimum degree stability of graphs forbidding some odd cycles
数学公众报告(120周年校庆系列第96场)
报告题目(Title)：Minimum degree stability of graphs forbidding some odd cycles
报告人(Speaker)：彭岳建 教授 (湖南大学)
地点(Place)：腾讯会议ID：374 426 505
时间(Time)：2023 年 4 月 28 日(周五), 15:00-16:00
邀请人(Inviter)：徐敏
报告摘要
We consider the minimum degree stability of graphs forbidding odd cycles: What is the tight bound on the minimum degree to guarantee that the structure of a C_{2k+1}-free graph inherits from the extremal graph (a balanced complete bipartite graph)? Andrásfai, Erdős and Sós showed that if a {C_3,C_5,…, C_{2k+1}}-free graph on n vertices has minimum degree greater than 2n/(2k+3), then it is bipartite. Häggkvist showed that for k∈{1,2,3,4}, if a C_{2k+1}-free graph on n vertices has minimum degree greater than 2n/(2k+3), then it is bipartite. Häggkvist also pointed out that this result cannot be extended to k≥5. In this paper, we give a complete answer for any k≥5. We show that if k≥5 and G is an n-vertex C_{2k+1}-free graph with δ(G)≥n/6+1, then G is bipartite, and the bound n/6+1 is tight. Furthermore, the result can be strengthened as follows. Let 2≤l≤k and n≥1000k^8 be integers. Let G be an n-vertex {C_3,C_5,…, C_{2l-1}; C_{2k+1}}-free graph. Then the following holds. (i) If l>(2k-1)/8 and δ(G)≥2n/(2k+3)+1, then G is bipartite, and the bound 2n/(2k+3)+1 is tight. If δ(G)=2n/(2k+3), then G must be a balanced blow up of C_{2k+3}. (ii) If l<(2k-1)/8 and δ(G)≥n/(2(2l+1))+1, then G is bipartite, and the bound n/(2(2l+1))+1 is tight. If δ(G)=n/(2(2l+1)), then G must be a graph taking 2l+1 vertex-disjoint copies of K_{n/(2(2l+1)), n/(2(2l+1))}, select a vertex in each of them such that these vertices form a cycle of length 2l+1. This is a joint work with Xiaoli Yuan.
主讲人简介
彭岳建，湖南大学数学学院教授，博士生导师。2001 年获 Emory 大学（美国）数学博士。2002-2012年在美国印第安纳州立大学数学系工作并获终身教授，2013年入选湖南省百人计划全职到湖南大学数学学院工作。主要研究方向为极值组合，在 JCTB, JCTA, CPC，SIDA 等知名期刊发表论文多篇，主持国家自然科学基金面上项目和重点项目。任中国数学会图论与组合分会委员，曾任中国数学会理事和中国工业与应用数学学会理事。
