On the Ramsey numbers for the tree graphs versus certain generalised wheel graphs

Given two simple graphs G and H, the Ramsey number R(G,H) is the smallest integer n such that for any graph of order n, either it contains G or its complement contains H. Let Tn be a tree graph of order n and Ws,m be the generalised wheel graph Ks+Cm. In this paper, we show that for n≥5,s≥2, R(Tn,Ws,6)=(s+1)(n−1)+1 and for n≥5,s≥1, R(Tn,Ws,7)=(s+2)(n−1)+1. We also determine the exact value of R(Tn,Ws,m) for large n and s.