Speaker：Bing Wei (University of Mississippi)

Time：2023-12-25 16:00

Location：Conference Room 686 at the 6th floor of Shuli Building at Haiyun Campus

Abstract:

A Gallai coloring of a complete graph is an edge-coloring such that no triangle has all its edges colored differently. A Gallai $k$-coloring is a Gallai coloring that uses $k$ colors. Given a graph $H$ and an integer $k\ge1$, the Gallai-Ramsey number $GR_k(H)$ is defined to be the minimum integer $n$ such that every Gallai $k$-coloring of the edges of $K_n$ contains a monochromatic copy of $H$. If $k=2$, $GR_2(G)$ is the classical Ramsey number, which means that Gallai-Ramsey number is a generalization of Ramsey number. In this talk, we present some of our recent results on the upper and lower bounds of Ramsey numbers and Gallai-Ramsey numbers for graphs with small chromatic numbers such as $\widehat{K}_m$ for $m\ge2$, where $\widehat{K}_m$ is a kipas with $m+1$ vertices obtained from the join of $K_1$ and a path $P_m$, $W_n$ (a wheel with $n+1$ vertices) obtained  from the join of  $K_1$ and a cycle $C_n$, and some graphs with small number of vertices. Exact values of Gallai-Ramsey numbers for some special graphs will also be provided. Our outcomes generalize several recent results which are obtained individually in different published papers.