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 = n0 (c) and k = 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 ε (0 < ε < 1), there exists n0 = n0 (c, ε) such that if n ≥ n0 then G contains a cycle Ct for all t with

Cycles and paths in graphs with large minimal degree

Let G be a simple graph of order n and minimal degree \u3e cn (0 \u3c c \u3c 1/2). We prove that (1) There exist n0 = n0(c) and k = k(c) such that if n \u3e n0 and G contains a cycle Ct for some t \u3e 2k, then G contains a cycle Ct-2s for some positive s \u3c k; (2) Let G be 2-connected and non-bipartite. For each ε (0 \u3c ε \u3c 1), there exists n0 = n0(c, ε) such that if n ≥ n0 then G contains a cycle Ct for all t with [2/c] - 2 ≤ t ≤ 2(1 - ε)cn. This answers positively a question of Erdos, Faudree, Gyárfás and Schelp. © 2004 Wiley Periodicals, Inc