On Ramsey Numbers for Trees Versus Wheels of Five or Six Vertices E.T. Baskoro 1, *, Surahmat 1 , S.M. Nababan 1 , and M. Miller 2 1 Department of Mathematics, Institut Teknologi Bandung, Jalan Ganesa 10 Bandung, Indonesia. e-mails: febaskoro, kana_s, nababang@dns.math.itb.ac.id 2 Department of Computer Science and Software Engineering, The University of Newcastle, Callaghan, NSW 2308, Australia. e-mail: mirka@cs.newcastle.edu.au Abstract. For given two graphs G dan H , the Ramsey number RðG; H Þ is the smallest positive integer n such that every graph F of order n must contain G or the complement of F must contain H . In [12], the Ramsey numbers for the combination between a star S n and a wheel W m for m ¼ 4; 5 were shown, namely, RðS n ; W 4 Þ¼ 2n  1 for odd n and n  3, otherwise RðS n ; W 4 Þ¼ 2n þ 1, and RðS n ; W 5 Þ¼ 3n  2 for n  3. In this paper, we shall study the Ramsey number RðG; W m Þ for G any tree T n . We show that if T n is not a star then the Ramsey number RðT n ; W 4 Þ¼ 2n  1 for n  4 and RðT n ; W 5 Þ¼ 3n  2 for n  3. We also list some open problems. 1. Introduction Let G be a graph with vertex set V ðGÞ and edge set EðGÞ. We define jGj¼jV ðGÞj. The graph G is the complement of graph G. For any set S  V ðGÞ, the induced subgraph by S is the maximal subgraph of G with vertex set S ; it is denoted by G½S . For x 2 V ðGÞ and B  V ðGÞ, define N B ðxÞ¼fy 2 B : xy 2 EðGÞg. Thus, the degree dðxÞ of a vertex x is jN V ðxÞj. A set S  V ðGÞ is said to be independent if any two vertices of S are not adjacent. Similarly, two edges e and f are independent if no endpoint of e is adjacent to either endpoint of f . Throughout the paper, C n denotes a cycle with n vertices. Let H n denote a cocktail-party graph, namely a graph obtained from a complete graph of 2n vertices by removing n independent edges. Let S n and T n denote a star and a tree of n vertices respectively. A wheel of n þ 1 vertices will be denoted by W n . Given two graphs G and H , define G þ H as a graph with vertex set V ðGÞ[ V ðH Þ and edge set EðGÞ[ EðH Þ[fxy : x 2 V ðGÞ; y 2 V ðH Þg. For * This work was supported by the QUE Project, Department of Mathematics ITB Indonesia Graphs and Combinatorics (2002) 18:717–721 Graphs and Combinatorics Ó Springer-Verlag 2002