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 be a star of order n and W_s,m be a generalised wheel K_s∨ 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 -good, W_s,5 -good, W_s,6 -good, and W_s,7 -good for s⩾ 2. In this paper, we study the Ramsey numbers R(S_n, W_s,8), and our results indicate that trees are not always W_s,8 -good.