The Ramsey numbers for disjoint unions of cycles Original Research Article

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. We shall establish the values of r(aC5, bC5) and r(aC7, bC7) almost precisely (where nG is the graph consisting of n vertex disjoint copies of G) extending the work of Mizuno and Sato, who proved similar results about r(aC4, bC4). Our technique also allows us to find a general upper bound for the Ramsey number r(aCn, aCm) for any a ⩾ 1, n, m ⩾ 3.