All Ramsey critical graphs for a large tree versus $t{K}_m$

Abstract

Let $H,H_1$ and $H_2$ be graphs, and let $H\to (H_1,H_2)$ denote that any red-blue coloring of $E(H)$ yields a red copy of $H_1$ or a blue copy of $H_2$. The Ramsey number for $H_1$ versus $H_2$, $r(H_1,H_2)$, is the minimum integer $N$ such that $K_N\to (H_1,H_2)$. The Ramsey critical graph $H$ for $H_1$ versus $H_2$ is a red-blue edge-colored $K_{N-1}$ such that $H\nrightarrow (H_1,H_2)$, where $N=r(H_1,H_2)$. In this paper, we characterize all Ramsey critical graphs for a large tree versus $t{K}_m$. As a corollary, we determine the star-critical Ramsey number for a large tree versus $t{K}_m$.