On the Star-Critical Ramsey Number of a Forest Versus Complete Graphs

Abstract

Let G and $G_1, G_2, \ldots , G_t$ be given graphs. By $G\rightarrow (G_1, G_2, \ldots , G_t)$, we mean if the edges of G are arbitrarily colored by t colors, then for some i, $1\le i\le t$, the spanning subgraph of G whose edges are colored with the i-th color, contains a copy of $G_i$. The Ramsey number $R(G_1, G_2, \ldots , G_t)$ is the smallest positive integer n such that $K_n\rightarrow (G_1, G_2, \ldots , G_t)$, and the size Ramsey number ${\hat{R}}(G_1, G_2, \ldots , G_t)$ is defined as $\min \{|E(G)|:~G\rightarrow (G_1, G_2, \ldots , G_t)\}$. Also, for given graphs $G_1, G_2, \ldots , G_t$ with $r=R(G_1, G_2, \ldots , G_t)$, the star-critical Ramsey number $R_*(G_1, G_2, \ldots , G_t)$ is defined as $\min \{\delta (G):~G\subseteq K_r, ~G\rightarrow (G_1, G_2, \ldots , G_t)\}$. In this paper, the Ramsey number and also the star-critical Ramsey number of a forest versus any number of complete graphs will be computed exactly in terms of the Ramsey number of the complete graphs. As a result, the computed star-critical Ramsey number is used to give a tight bound for the size Ramsey number of a forest versus a complete graph.