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Arithmetic Progressions of Cycle Lengths in Graphs

Jacques Verstraete
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Abstract

A recently posed question of Haggkvist and Scott's asked whether or not there exists a constant c such that if G is a graph of minimum degree ck then G contains cycles of k consecutive even lengths. In this paper we answer the question by proving that for k > 1, a bipartite graph of average degree at least 4k and girth g contains cycles of (g/2-1)k consecutive even lengths. We also obtain a short proof of the theorem of Bondy and Simonovits, that a graph of order n and size at least 8(k-1)n^{1 + 1/k} has a cycle of length 2k.

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Jacques Verstraete. 2002-04-17. Arithmetic Progressions of Cycle Lengths in Graphs. https://arxiv.org/abs/math/0204222

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