The Ramsey numbers R(Tn, W6) for Delta (Tn) geq n-3

K e y w o r d s R a m s e y number, Tree, Wheel. 1. I N T R O D U C T I O N All graphs considered in this paper are finite simple graphs without loops. For two given graphs G1 and G2, the Ramsey number R(G1, G2) is the smallest positive integer n such that for any graph G of order n, either G contains G1 or G contains G2, where G is the complement of G. Let G be a graph. We use V(G) and E(G) to denote its vertex set and edge set, respectively. The neighborhood N(v) of a vertex v is the set of vertices adjacent to v in G. The minimum and maximum degree of G are denoted by 5(G) and A(G), respectively. For a vertex v e V(G) and a subgraph H of G, NH(V) is the set of neighbors of v contained in H, i.e., NH(V) = N(v) N V(H). We let dH(v) = INH(v)]. For S C V(G), G[S] denotes the subgraph induced by S in G. Let U, V be two disjoint vertex sets. We use E(U, V) to denote the set of edges between U and V. Let m be a positive integer. We use mG to denote m vertex disjoint copies of G. A path and a cycle of order n are denoted by Pn and C~, respectively. A star S,~ (n > 3) is a bipart i te graph Kl ,n-1 . We use K3,3,...,3 to denote a balanced complete n /3-par t i t e graph of order n = 0 (rood 3). A wheel Wn = {x} + C~ is a graph of n + 1 vertices, tha t is, a vertex x, called the hub of the wheel, adjacent to all vertices of C~. S,(l, m) is a tree of order n obtained from S~-t×m by subdividing each of l chosen edges m times. Sn(1) is a tree of order n obtained from an St and an Snl by adding an edge joining the centers of them. A graph on n vertices is pancycIic if it contains cycles of every length l, 3 < 1 < n. Many thanks to the anonymous referees for their many helpful comments and suggestions, especially the simple proof of Theorem 1, which have considerably improved the presentation of the paper. This project was supported by NSFC. *This project was supported by Nanjing University Talent Development Foundation. 0893-9659/04/$ see front matter (~) 2004 Elsevier Ltd. All rights reserved. doi: 10.1016/S0893-9659 (04)00008-4 Typeset by ~4j~S-TEX