The Ramsey numbers for disjoint unions of cycles

As usual, for simple graphs G and H, let the Ramsey number r(G,H) be defined as the least number n such that for any graph K of order n, either G is a subgraph of K or H is a subgraph of (K) over bar. We shall establish the values of r(aC(5),bC(5)) and r(aC(7),bC(7)) almost precisely (where nG is the graph consisting of n vertex disjoint copies of G) extending the work of Mizuno and Sate, who proved similar results about r(aC(4),bC(4)). Our technique also allows us to find a general upper bound for the Ramsey number r(aC(n), aC(m)) for any a greater than or equal to 1,n,m greater than or equal to 3.