The Ramsey Numbers of Paths Versus Wheels: a Complete Solution

Let G1 and G2 be two given graphs. The Ramsey number R(G1, G2) is the least integer r such that for every graph G on r vertices, either G contains a G1 or G contains a G2. We denote by Pn the path on n vertices and Wm the wheel on m + 1 vertices. Chen et al. and Zhang determined the values of R(Pn,Wm) when m 6 n + 1 and when n + 2 6 m 6 2n, respectively. In this paper we determine all the values of R(Pn,Wm) for the left case m > 2n + 1. Together with Chen et al.’s and Zhang’s results, we give a complete solution to the problem of determining the Ramsey numbers of paths versus wheels.