Long alternating cycles in edge-colored complete graphs

  • Authors:

  • Hao Li;Guanghui Wang;Shan Zhou

  • Affiliations:

  • Laboratoire de Recherche en Informatique, UMR, C.N.R.S, Université de Paris-sud, Orsay cedex, France and School of Mathematics and Statistics, Lanzhou University, Lanzhou, China;Laboratoire de Recherche en Informatique, UMR, C.N.R.S, Université de Paris-sud, Orsay cedex, France and School of Mathematics and System Science, Shandong University, Jinan, Shandong, China;Laboratoire de Recherche en Informatique, UMR, C.N.R.S, Université de Paris-sud, Orsay cedex, France and School of Mathematics and Statistics, Lanzhou University, Lanzhou, China

  • Venue:
  • FAW'07 Proceedings of the 1st annual international conference on Frontiers in algorithmics
  • Year:
  • 2007

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Abstract

Let Knc denote a complete graph on n vertices whose edges are colored in an arbitrary way. And let Δ(Knc) denote the maximum number of edges of the same color incident with a vertex of Knc. A properly colored cycle (path) in Knc, that is, a cycle (path) in which adjacent edges have distinct colors is called an alternating cycle (path). Our note is inspired by the following conjecture by B. bollobás and P. Erdös(1976): If Δ(Knc) n/2⌉, then Knc contains an alternating Hamiltonian cycle. We prove that if Δ(Knc) n/2⌉, then Knc contains an alternating cycle with length at least ⌊n+2/3⌋ + 1.