The Ramsey numbers of paths versus wheels

For two given graphs G1G1 and G2G2, the Ramsey number R(G1,G2)R(G1,G2) is the smallest integer n such that for any graph G of order n, either G contains G1G1 or the complement of G contains G2G2. Let PnPn denote a path of order n and WmWm a wheel of order m+1m+1. In this paper, we show that R(Pn,Wm)=2n-1R(Pn,Wm)=2n-1 for m even and n⩾m-1⩾3n⩾m-1⩾3 and R(Pn,Wm)=3n-2R(Pn,Wm)=3n-2 for m odd and n⩾m-1⩾2n⩾m-1⩾2.