The Ramsey Numbers R(Cm, K7) and R(C7, K8)

is the smallest integer n such that for any graph G of order n, either G contains G1 or the complement of G contains G2. Let Cm denote a cycle of length m and Kn a complete graph of order n. In this paper we show that R(Cm,K7) = 6m − 5 for m ≥ 7 and R(C7,K8) = 43, with the former result confirms a conjecture due to Erdös, Faudree, Rousseau and Schelp that R(Cm,Kn) = (m − 1)(n − 1) + 1 for m ≥ n ≥ 3 and (m,n) 6 = (3, 3) in the case where n = 7