Some Tree-book Ramsey Numbers

Abstract

For two given graphs $G_1$ and $G_2$, the Ramsey number $R(G_1,G_2)$ is the smallest integer $x$ such that for any graph $G$ of order $n$, either $G$ contains $G_1$ or the complement of $G$ contains $G_2$. In this paper, we study a large class of trees $T$ as studied by Cockayne in [3], including paths and trees which have a vertex of degree one adjacent to a vertex of degree two, as special cases. We evaluate some $R(T_m^{\prime },B_m)$, where $T_n^{\prime }\in \mathbb{T}$ and $B_m$ is a book of order $m+2$. Besides, some bounds for $R(T_n^{\prime },B_n)$ are obtained.