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A Note on Coloring \((4K_1, C_4, C_6)\)-Free Graphs with a \(C_7\)

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Abstract

Even-hole-free graphs are a graph class of much interest. Foley et al. (Graphs Comb. 36(1):125–138, 2020) have recently studied \((4K_1, C_4, C_6)\)-free graphs, which form a subclass of even-hole-free graphs. Specifically, Foley et al. have shown an algorithm for coloring these graphs via bounded clique-width if they contain a \(C_7\). In this note, we give a simpler and much faster algorithm via a more restrictive graph parameter, neighborhood diversity.

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Acknowledgements

Many thanks to Irena Penev for helpful discussions and feedback on a draft of this note. M. Koutecký was partially supported by Charles University project UNCE/SCI/004 and by the project 19-27871X of the Czech Science Foundation (Grantová Agentura ČR).

Funding

This study was funded by Grantová Agentura Ceské Republiky (Grant no. 19-27871X) and Univerzita Karlova v Praze (Grant no. UNCE/SCI/004).

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  1. Computer Science Institute, Faculty of Mathematics and Physics, Charles University, Malostranské náměstí 25, Prague 1, Czech Republic

    Martin Koutecký

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Correspondence to Martin Koutecký.

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Koutecký, M. A Note on Coloring \((4K_1, C_4, C_6)\)-Free Graphs with a \(C_7\). Graphs and Combinatorics 38, 149 (2022). https://doi.org/10.1007/s00373-022-02553-4

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  • DOI: https://doi.org/10.1007/s00373-022-02553-4

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