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 $i$, $1\leq i\leq t$, 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 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.