Ramsey Numbers of Stars Versus Generalised Wheels

For two graphs G and H,  the Ramsey number R(G, H) is the smallest integer n such that for any n-vertex graph, either it contains G or its complement contains H. Let \(S_{n}\) $S_n$ be a star of order n and \(W_{s,m}\) $W_{s,m}$ be a generalised wheel \(K_{s}\vee C_{m}.\) $K_s\vee C_m.$ Previous studies by Wang and Chen (Graphs Comb 35(1):189–193, 2019) and Chng et al. (Discret Math 344(8):112440, 2021) imply that a tree is \(W_{s,4}\) $W_{s,4}$ -good, \(W_{s,5}\) $W_{s,5}$ -good, \(W_{s,6}\) $W_{s,6}$ -good, and \(W_{s,7}\) $W_{s,7}$ -good for \(s\geqslant 2.\) $s⩾2.$ In this paper, we study the Ramsey numbers \(R(S_{n}, W_{s,8}),\) $R(S_n,W_{s,8}),$ and our results indicate that trees are not always \(W_{s,8}\) $W_{s,8}$ -good.