Title：A Ramsey--Turan theory for tilings in graphs

Abstract：Given an integer $rge 2$ 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 ErdH{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).

Speaker：Donglei Yan, Shandong University, mainly works on graph coloring, extremal graph theory and graph partition. His results are mainly published on Journal of Graph Theory, European Journal of Combinatorics, Discrete Mathematics etc..

Date：2:00pm-5:00pm 2021-5-13日（Thursday）.

Tencent Meeting ID： 832 786 786

Organizer：School of Mathematical Science

Students and teachers who are interested in graph theory are welcome.