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Cut-Edges and Regular Factors in Regular Graphs of Odd Degree

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Abstract

We study 2k-factors in \((2r+1)\)-regular graphs. Hanson, Loten, and Toft proved that every \((2r+1)\)-regular graph with at most 2r cut-edges has a 2-factor. We generalize their result by proving for \(k\le (2r+1)/3\) that every \((2r+1)\)-regular graph with at most \(2r-3(k-1)\) cut-edges has a 2k-factor. Both the restriction on k and the restriction on the number of cut-edges are sharp. We characterize the graphs that have exactly \(2r-3(k-1)+1\) cut-edges but no 2k-factor. For \(k>(2r+1)/3\), there are graphs without cut-edges that have no 2k-factor, as studied by Bollobás, Saito, and Wormald.

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Funding

Alexandr V. Kostochka: Research supported by National Science Foundation grant DMS1600592 and grants 18-01-00353A and 19-01-00682 of the Russian Foundation for Basic Research. Douglas B. West: Research supported by National Natural Science Foundation of China grants NSFC 11871439 and 11971429. Dara Zirlin: Research supported by Arnold O. Beckman Campus Research Board Award RB20003 of the University of Illinois at Urbana-Champaign.

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Authors and Affiliations

  1. University of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA

    Alexandr V. Kostochka, Douglas B. West & Dara Zirlin

  2. Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia

    Alexandr V. Kostochka

  3. Université de Bordeaux, LaBRI UMR 5800, 33400, Talence, France

    André Raspaud

  4. University of Southern Denmark, Odense, Denmark

    Bjarne Toft

  5. Zhejiang Normal University, Jinhua, 321004, China

    Douglas B. West

Authors
  1. Alexandr V. Kostochka
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  2. André Raspaud
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  3. Bjarne Toft
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  4. Douglas B. West
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  5. Dara Zirlin
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Correspondence to Douglas B. West.

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Kostochka, A.V., Raspaud, A., Toft, B. et al. Cut-Edges and Regular Factors in Regular Graphs of Odd Degree. Graphs and Combinatorics 37, 199–207 (2021). https://doi.org/10.1007/s00373-020-02242-0

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  • DOI: https://doi.org/10.1007/s00373-020-02242-0

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