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 \(R(T’_m, B_m)\), where \(T’_n \in \mathbb{T}\) and \(B_m\) is a book of order \(m+2\). Besides, some bounds for \(R(T’_n, B_n)\) are obtained.