The Ramsey Numbers of Trees Versus Generalized Wheels

Abstract

For two given graphs $G_1$ and $G_2$, the Ramsey number $R(G_1,G_2)$ is the smallest integer n such that for any graph G of order n, either G contains $G_1$ or its complement ${\overline{G}}$ contains $G_2$. Let $P_n, S_n$ and $T_n$ denote a path, a star and a tree of order n, respectively. A generalized wheel, denoted by $W_{s,m}$, is the join of a complete graph $K_s$ and a cycle $C_m$. In this paper, we show that $R(T_n,W_{s,4})=(n-1)(s+1)+1$ for $n\ge 3,s\ge 2$ and $R(T_n,W_{s,5})=(n-1)(s+2)+1$ for $n\ge 3,s\ge 1$. These generalize some known results on Ramsey numbers for a tree versus a wheel.