A refinement of a result of Corrádi and Hajnal

Combinatorica volume 35, pages497–512(2015)Cite this article

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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Affiliations

  1. School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ, 85287, USA

    H. A. Kierstead

  2. Department of Mathematics, University of Illinois, Urbana, IL, 61801, USA

    A. V. Kostochka

  3. Institute of Mathematics, Novosibirsk, Russia

    A. V. Kostochka

Authors
  1. H. A. Kierstead
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  2. A. V. Kostochka
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Correspondence to H. A. Kierstead.

Additional information

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