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 n0 1⁄4 n0(c) and k 1⁄4 k(c) such that if n > n0 and G contains a cycle Ct for some t > 2k ; then G contains a cycle Ct 2s for some positive s < k ; (2) Let G be 2-connected and nonbipartite. For each " (0 < " < 1), there exists n0 1⁄4 n0(c; ") such that if n n0 then G contains a cycle Ct for all t with d2ce 2 t 2(1 ")cn: This answers positively a question of Erdős, Faudree, Gyárfás and Schelp. 2004 Wiley Periodicals, Inc. J Graph Theory 47: 39–52, 2004