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Vizing Bound for the Chromatic Number on Some Graph Classes

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Abstract

We are interested in hereditary classes of graphs \({\mathcal {G}}\) such that every graph \(G \in {\mathcal {G}}\) satisfies \(\varvec{\chi }(G) \le \omega (G) + 1\), where \(\chi (G)\) (\(\omega (G)\)) denote the chromatic (clique) number of G. This upper bound is called the Vizing bound for the chromatic number. Apart from perfect graphs few classes are known to satisfy the Vizing bound in the literature. We show that if G is (\(P_6, S_{1, 2, 2}\), diamond)-free, then \(\chi (G) \le \omega (G)+1\), and we give examples to show that the bound is sharp.

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Acknowledgments

The authors thank the anonymous referees for their suggestions and comments.

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Authors and Affiliations

  1. Computer Science Unit, Indian Statistical Institute, Chennai Centre, Chennai, 600 113, India

    T. Karthick

  2. CNRS, Laboratoire G-SCOP, University of Grenoble, Grenoble, France

    Frédéric Maffray

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  1. T. Karthick
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  2. Frédéric Maffray
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Correspondence to Frédéric Maffray.

Additional information

Frédéric Maffray: Partially supported by ANR project STINT under reference ANR-13-BS02-0007.

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Karthick, T., Maffray, F. Vizing Bound for the Chromatic Number on Some Graph Classes. Graphs and Combinatorics 32, 1447–1460 (2016). https://doi.org/10.1007/s00373-015-1651-1

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  • DOI: https://doi.org/10.1007/s00373-015-1651-1

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