Cycle lengths and minimum degree of graphs

Chun Hung Liu, Jie Ma

Research output: Contribution to journalArticle

4 Scopus citations

Abstract

There has been extensive research on cycle lengths in graphs with large minimum degree. In this paper, we obtain several results that are tight in this area. Let G be a graph with minimum degree at least k+1. We prove that if G is bipartite, then there are k cycles in G whose lengths form an arithmetic progression with common difference two. For a general graph G, we show that G contains ⌊k/2⌋ cycles with consecutive even lengths and k−3 cycles whose lengths form an arithmetic progression with common difference one or two. In addition, if G is 2-connected and non-bipartite, then G contains ⌊k/2⌋ cycles with consecutive odd lengths. Thomassen (1983) made two conjectures on cycle lengths modulo a fixed integer k: (1) every graph with minimum degree at least k+1 contains cycles of all even lengths modulo k; (2) every 2-connected non-bipartite graph with minimum degree at least k+1 contains cycles of all lengths modulo k. These two conjectures, if true, are best possible. Our results confirm both conjectures when k is even. When k is odd, we show that minimum degree at least k+4 suffices. This improves all previous results in this direction. Moreover, our results derive new upper bounds of the chromatic number in terms of the longest sequence of cycles with consecutive (even or odd) lengths.

Original languageEnglish (US)
Pages (from-to)66-95
Number of pages30
JournalJournal of Combinatorial Theory. Series B
Volume128
DOIs
StatePublished - Jan 2018

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics

Keywords

  • Cycle length modulo k
  • Cycles of consecutive lengths
  • Minimum degree
  • Thomassen's conjecture

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