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

Let G and G(1),G(2),....,G(t) be given graphs. By G -> (G(1),G(2),...,G(t)), we mean if the edges of G are arbitrarily colored by t colors, then for some i, 1 <= i <= 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),...,G(t)) is the smallest positive integer n such that K-n -> (G(1),G(2),...,G(t)), and the size Ramsey number R(G(1),G(2)...,G(t)) is defined as min{vertical bar E(G)vertical bar: G -> (G(1),G(2),...,G(t))}. Also, for given graphs G(1),G(2),...,G(t) with r=R(G(1),G(2),...,G(t)), the star-critical Ramsey number R*(G(1),G(2,)...,G(t)) is defined as min{delta(G): G I K-r, G -> (G(1),G(2),...,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.