The cycle-complete graph Ramsey number r(C 8, K 8)

Abstract

The cycle-complete graph Ramsey number r(C m, K n) is the smallest integer N such that every graph G of order N contains a cycle C m on m vertices or has independent number α(G) ≥ n. It has been conjectured by Erdös, Faudree, Rousseau and Schelp that r(C m, K n) = (m - 1)(n - 1) + 1 for all m ≥ n ≥ 3 (except r(C 3, K 3) = 6). In this paper we will present a proof for the conjecture in the case n = m = 8.