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  > 2 k , then G contains a cycle C t  − 2 s for some positive s  <  k ; (2) Let G be 2‐connected and nonbipartite. For each ε(0 < ε < 1), there exists n 0  =  n 0 ( c ,ε) such that if n  ≥  n 0 then G contains a cycle C t for all t with $\left \lceil { 2\over c}\right \rceil -2\leq t\leq 2(1-\varepsilon) cn$ . This answers positively a question of Erdős, Faudree, Gyárfás and Schelp. © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 39–52, 2004