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). [References: 6]
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