On Ramsey Numbers for Trees Versus Wheels of Five or Six Vertices

 For given two graphs G dan H, the Ramsey numberR(G,H) is the smallest positive integer n such that every graph F of order n must contain G or the complement of F must contain H. In [12], the Ramsey numbers for the combination between a star Sn and a wheel Wm for m=4,5 were shown, namely, R(Sn,W4)=2n−1 for odd n and n≥3, otherwise R(Sn,W4)=2n+1, and R(Sn,W5)=3n−2 for n≥3. In this paper, we shall study the Ramsey number R(G,Wm) for G any tree Tn. We show that if Tn is not a star then the Ramsey number R(Tn,W4)=2n−1 for n≥4 and R(Tn,W5)=3n−2 for n≥3. We also list some open problems.