2002/04/17 by Jacques Verstraete
Mathematics · #math.CO #msc:05Cxx
published as Combinatorics, Probability and Computing 9 (2000) No 4 369-373
arxiv created 2002/04/17 · arxiv updated 2009/11/30
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)n1 + 1/k has a cycle of length 2k.