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.