Cycles and paths in graphs with large minimal degree

Let G be a simple graph of order n and minimal degree > cn (0 < c < 1/2). We prove that (1) There exist n(0) = n(0)(c) and k = k(c) such that if n > n(0) and G contains a cycle C-t for some t > 2k, then G contains a cycle Ct-2s for some positive s < k; (2) Let G be 2-connected and non-bipartite. For each E (0 < E < 1), there exists no = no(C, E) such that if n > no then G contains a cycle C-t for all t with [(2)/(c)] - 2 less than or equal to t less than or equal to 2(1 - epsilon)cn. This answers positively a question of Erdos, Faudree, Gyarfas and Schelp. (C) 2004 Wiley Periodicals, Inc.