The Ramsey numbers for disjoint unions of trees

For given graphs G and H, the Ramsey number R(G,H) is the smallest natural number n such that for every graph F of order n: either F contains G or the complement of F contains H. In this paper we investigate the Ramsey number R(∪G,H), where G is a tree and H is a wheel Wm or a complete graph Km. We show that if n ≥ 3, then R(kSn, W4) = (k+1)n for k ≥ 2, even n and R(kSn,W4) = (k+1)n−1 for k ≥ 1 and odd n. We also show that R(ki=1 Tni ,Km) = R(Tnk ,Km)+ ∑k−1 i=1 ni.