On arithmetic progressions of cycle lengths in graphs

A question recently posed by Haggkvist and Scott 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 greater than or equal to 2, 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 Bendy 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.