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Long Alternating Cycles in Edge-Colored Complete Graphs

  • Hao Li
  • Guanghui Wang
  • Shan Zhou
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4613)

Abstract

Let \(K_n^c\) denote a complete graph on n vertices whose edges are colored in an arbitrary way. And let \(\Delta (K_n^c)\) denote the maximum number of edges of the same color incident with a vertex of \(K_n^c\). A properly colored cycle (path) in \(K_n^c\), 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. Erdős(1976): If \(\Delta(K_n^c)<\lfloor n/2\rfloor\), then \(K_n^c\) contains an alternating Hamiltonian cycle. We prove that if \(\Delta (K_n^c)< \lfloor n/2\rfloor\), then \(K_n^c\) contains an alternating cycle with length at least \(\lceil \frac{n+2}{3}\rceil+1\).

Keywords

alternating cycle color degree edge-colored graph 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Hao Li
    • 1
    • 3
  • Guanghui Wang
    • 1
    • 2
  • Shan Zhou
    • 1
    • 3
  1. 1.Laboratoire de Recherche en Informatique, UMR 8623, C.N.R.S-Université de Paris-sud, 91405-Orsay cedexFrance
  2. 2.School of Mathematics and System Science, Shandong University, 250100 Jinan, ShandongChina
  3. 3.School of Mathematics and Statistics, Lanzhou University, 730000 LanzhouChina

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