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A Note on \(\Delta \)-Critical Graphs

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Abstract

A k-critical graph is a k-chromatic graph whose proper subgraphs are all \((k-1)\)-colourable. An old open problem due to Borodin and Kostochka asserts that for \(k\ge 9\), no k-critical graph G with \(k=\Delta (G)\) exists, where \(\Delta (G)\) denotes the maximum degree of G. We show that if a certain special list-colouring property holds for every 8-critical graph with \(\Delta =8\) (which is true for the apparently only known example), then the Borodin–Kostochka Conjecture holds. We also briefly survey constructions of \(\Delta \)-critical graphs with \(\Delta \le 8\), highlighting the apparent scarcity of such graphs once \(\Delta \) exceeds 6.

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Acknowledgements

The first author is grateful to Université Paris Cité for a professeur invité position (2019) held at IRIF, during which this work was done. The authors also thank the anonymous referees for their helpful comments.

Funding

This work is supported by the ANR project HOSIGRA (ANR-17-CE40-0022) and by NSERC-Canada.

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Author notes

  1. Penny Haxell and Reza Naserasr have equally contributed to this paper.

Authors and Affiliations

  1. Department of Combinatorics and Optimization, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

    Penny Haxell

  2. IRIF, Université Paris Cité, CNRS, 75013, Paris, France

    Reza Naserasr

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  1. Penny Haxell
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  2. Reza Naserasr
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Correspondence to Penny Haxell or Reza Naserasr.

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Haxell, P., Naserasr, R. A Note on \(\Delta \)-Critical Graphs. Graphs and Combinatorics 39, 101 (2023). https://doi.org/10.1007/s00373-023-02696-y

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