On a ramsey-type problem of J. A. Bondy and P. Erdös. II

AbstractLet R(Cn, Cp) be the smallest integer m for which the following statement is true: If a G graph has at least m vertices, then either G contains a Cn (cycle of length n) or G contains a Cp.Bondy and Erdös showed in [1] that R(Cn, Cn) = 2n − 1 if n odd, n > 3; in [2] we showed that R(Cn, Cp) = 2n − 1 if 3 ≤ p < n, p odd, n > 4. In this paper we finish the investigation of R(Cn, Cp) by showing: Theorem 1. Let n even n ≥ 7, 3 ≤r ≤ n2, then R(Cn, C2r) = n + r − 1.Theorem 2. Let n odd n ≥ 7, 3 ≤ r ≤ (n − 1)2, then R(Cn, C2r) = max{+ r − 1, 4r − ;