The Ramsey numbers of wheels versus odd cycles

Given two 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 contains G(2). Let C-m denote a cycle of order m and W-n a wheel of order n + 1. In this paper, it is shown that R(W-n, C-m) = 2n + 1 for m odd, n >= 3(m - 1)/2 and (m, n) not equal (3, 3), (3, 4), and R(W-n, C-m) = 3m - 2 for m, n odd and m < n < 3(m - 1)/2. (C) 2014 The Authors. Published by Elsevier B.V.