  Chen et al. (2004) strongly conjectured that R(Tn,Wm)=2n-1 if the maximum
degree of Tn is small and m is even. Related to the conjecture, it is
interesting to know for which tree Tn, we have R(Tn,Wm) > 2n-1 for even m. In
this paper, we find the Ramsey number R(Tn,W8) for tree Tn with the maximum
degree of Tn is at least n-3, namely R(Tn,W8) > 2n-1 for almost all such tree
Tn. We also prove that if the maximum degree of Tn is large, then R(Tn,Wm) is a
function of both m and n. In the end, we refine the conjecture of Chen et al.
by giving the condition of small maximum degree. For a tree Tn with large
maximum degree and even m, the R(Tn,Wm) is also unknown in general. In this
paper, we shall determine the Ramsey number R(Tn,W8) for all trees Tn of order
n with the maximum degree of Tn is at least n-3.
