On Arithmetic Progressions of Cycle Lengths in Graphs

  • Authors:

  • Jacques Verstraëte

  • Affiliations:

  • Department of Pure Mathematics and Mathematical Statistics, 16 Mill Lane, Cambridge CB2 1SB, England (e-mail: J.B.A.Verstraete@dpmms.cam.ac.uk)

  • Venue:
  • Combinatorics, Probability and Computing
  • Year:
  • 2000

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Abstract

A question recently posed by Häggkvist 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 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 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.