A refinement of a result of Corrádi and Hajnal

Henry Kierstead, A. V. Kostochka

Research output: Contribution to journalArticle

6 Scopus citations

Abstract

Corrádi and Hajnal proved that for every k ≥ 1 and n ≥ 3k, every n-vertex graph with minimum degree at least 2k contains k vertex-disjoint cycles. This implies that every 3k-vertex graph with maximum degree at most k − 1 has an equitable k-coloring. We prove that for s∈{3,4} if an sk-vertex graph G with maximum degree at most k has no equitable k-coloring, then G either contains K<inf>k+1</inf> or k is odd and G contains K<inf>k,k</inf>. This refines the above corollary of the Corrádi-Hajnal Theorem and also is a step toward the conjecture by Chen, Lih, and Wu that for r ≥ 3, the only connected graphs with maximum degree at most r that are not equitably r-colorable are K<inf>r,r</inf> (for odd r) and K<inf>r+1</inf>.

Original languageEnglish (US)
Pages (from-to)497-512
Number of pages16
JournalCombinatorica
Volume35
Issue number4
DOIs
StatePublished - Aug 5 2014

Keywords

  • 05C15
  • 05C35

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Computational Mathematics

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