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 Erdo&#x030B;s. By a different approach, we show in this paper that if G is a graph with minimum degree δ(G)⩾3k for any positive integer k, then G contains k+1 cycles C0, C1, …, Ck such that k+1<|E(C0)|<|E(C1)|<…<|E(Ck)|, |E(Ci)|− |E(Ci−1)|=2, i⩽i⩽k−1, and 1⩽|E(Ck)|−|E(Ck−1)|⩽2, and furthermore, if δ(G)⩾3k+1, then |E(Ck)|−|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⩾k+2.