The Ramsey Numbers for Disjoint Union of Stars

The Ramsey number for a graph G versus a graph H, denoted by R(G,H), is the smallest positive integer n such that for any graph F of order n, either F contains G as a subgraph or F contains H as a subgraph. In this paper, we investigate the Ramsey numbers for union of stars versus small cycle and small wheel. We show that if ni ≥ 3 for i = 1, 2, . . . , k and ni ≥ ni+1 ≥ √ ni − 2, then R( ∪k i=1 S1+ni , C4) = ∑k i=1 ni + k+1 for k ≥ 2. Furthermore, we show that if ni is odd and 2ni+1 ≥ ni for every i, then R( ∪k i=1 Sni ,W4) = R(Snk ,W4) + ∑k−1 i=1 ni for k ≥ 1.