The Ramsey numbers of paths versus wheels

For two given graphs G₁ and G₂, the Ramsey number R(G₁,G₂) is the smallest integer n such that for any graph G of order n, either G contains G₁ or the complement of G contains G₂. Let P_n denote a path of order n and W_m a wheel of order m+1. In this paper, we show that R(P_n,W_m)=2n-1 for m even and n?m-1?3 and R(P_n,W_m)=3n-2 for m odd and n?m-1?2.