Distribution of cycle lengths in graphs

Bondy and Vince proved that every graph with minimum degree at least three contains two cycles whose lengths differ by one or two, which answers a question raised by Erdos. By a different approach. we show in this paper that if G is a graph with minimum degree delta(G) greater than or equal to 3k for any positive integer k, then G contains k + 1 cycles C-0, C-1,..., C-k such that k+ 1 < \E(C-0)\ < \E(C-1)\ <... < \E(C-k)\, \E(C-1)\ - \E(Cl - 1)\ = 2. i less than or equal to i less than or equal to k - 1. and 1 less than or equal to \E(C-k)\ - \E(Ck-1)\ less than or equal to 2, and further-more, if delta(G) greater than or equal to 3(k+1), then \E(C-k)\ - \E(Ck-1)\ = 2, To settle a problem proposed by Bondy and Vince, we obtain that if G is a nonbipartite 3-connected graph with minimum degree at least 3k for any positive integer k. then G contains 2k cycles of consecutive lengths m, m+ 1, ..., m +2k - 1 for some integer m greater than or equal to k+2. (C) 2001 Elsevier Science (USA).