Minimum Degree and Even Cycle Lengths

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Bulletin of the Institute of Combinatorics and its Applications


A classic result of Dirac states that if G is a 2-connected graph of order n with minimum degree δ ≥ 3, then G contains a cycle of length at least min{n, 2δ}. In this paper, we consider the problem of determining the number of different odd or even cycle lengths that must exist under the minimum degree condition. We conjecture that there are δ − 1 even cycles of different lengths, and when G is nonbipartite, that there are δ− 1 odd cycles of different lengths. We prove this conjecture when δ = 3. Related results concerning the number of different even cycle lengths supporting the conjecture are also included. In particular, we show that there are always at least (δ − 1)/2 even cycles of different lengths.