A refinement of a result of Corrádi and Hajnal

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 k+1 or k is odd and G contains K k,k . 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 r,r (for odd r) and K r+1.

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Correspondence to H. A. Kierstead.

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Research of this author is supported in part by NSA grant H98230-12-1-0212.

Research of this author is supported in part by NSF grant DMS-0965587 and by grants 12-01-00448 and 12-01-00631 of the Russian Foundation for Basic Research.

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Kierstead, H.A., Kostochka, A.V. A refinement of a result of Corrádi and Hajnal. Combinatorica 35, 497–512 (2015). https://doi.org/10.1007/s00493-014-3059-6

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Mathematics Subject Classification (2000)

  • 05C15
  • 05C35