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On Bounding the Difference of the Maximum Degree and the Clique Number

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Abstract

For every \(k \in {\mathbb {N}}_0\), we consider graphs in which for any induced subgraph, \(\Delta \le \omega - 1 + k\) holds, where \(\Delta \) is the maximum degree and \(\omega \) is the maximum clique number of the subgraph. We give a finite forbidden induced subgraph characterization for every \(k\). As an application, we find some results on the chromatic number \(\chi \) of a graph. B. Reed stated the conjecture that for every graph, \(\chi \le \lceil \frac{\Delta + \omega + 1 }{2}\rceil \) holds. Since this inequality is fulfilled by graphs in which \(\Delta \le \omega + 2\) holds, our results provide a hereditary graph class for which the conjecture holds.

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Acknowledgments

We want to thank the reviewers for their many helpful comments.

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Authors and Affiliations

  1. Université Pierre et Marie Curie, Combinatoire et Optimisation, 4 place Jussieu, 75252, Paris, France

    Oliver Schaudt

  2. Institut für Informatik, Universität zu Köln, Weyertal 80, 50931, Cologne, Germany

    Vera Weil

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  1. Oliver Schaudt
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  2. Vera Weil
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Correspondence to Vera Weil.

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Schaudt, O., Weil, V. On Bounding the Difference of the Maximum Degree and the Clique Number. Graphs and Combinatorics 31, 1689–1702 (2015). https://doi.org/10.1007/s00373-014-1468-3

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  • DOI: https://doi.org/10.1007/s00373-014-1468-3

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