报告题目：A Ramsey--Turan theory for tilings in graphs

报告简介：Given an integer $r\ge 2$ and a graph $G$, let $\alpha_{r}(G)$ be the maximum size of a $K_{r}$-free subset of vertices and write $\alpha(G)=\alpha_2(G)$. A central topic in Ramsey--Tur\'{a}n theory, initiated by Erd\H{o}s and S\'{o}s, is to determine $RT_{r}(n;H; o(n))$, the minimum number of edges which guarantees that every $n$-vertex graph $G$ with $\alpha_{r}(G) =o(n)$ contains a copy of $H$. For a $k$-vertex graph $F$ and a graph $G$, an \emph{$F$-tiling} is a collection of vertex-disjoint copies of $F$ in $G$. We call an $F$-tiling \emph{perfect} if it covers the vertex set of $G$. We will also refer to a perfect $F$-tiling as an \emph{$F$-factor}, which is a fundamental object in graph theory with a wealth of results from various aspects. Motivated by Ramsey-Turán theory, a recent trend has focused on reducing the minimum degree condition forcing the existence of $F$-factors in graphs by adding an extra condition that provides pseudorandom properties.

In this talk, we mainly investigate the effect of imposing the condition that $\alpha_{r}(G)=o(n)$ by studying the minimum degree thresholds for $K_k$-tilings, and more generally, $F$-tilings. Similar questions for $F$-factors are considered where the condition $\alpha_{r}(G)=o(n)$ is replaced by $\alpha_{r}^*(G)=o(n)$ ( $r$-partite hole number).

报告人： 杨东雷，山东大学数据科学研究院，主要研究方向包括图的染色问题、极值问题以及有向图的划分问题。2018至2019年于佐治亚理工学院访问。目前已在Journal of Graph Theory, European Journal of Combinatorics, Discrete Mathematics等国际期刊发表论文多篇

报告时间：2021年5月13日（星期四）下午 2:00-5:00.

报告地点：腾讯会议，ID： 832 786 786

主办单位：扬州大学数学科学学院

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