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On the Chromatic Number of (\(P_6\), Diamond)-Free Graphs

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Abstract

In this paper, we first show that every (\(P_6\), diamond, \(K_4\))-free graph is 6-colorable. We also give an example of a (\(P_6\), diamond, \(K_4\))-free graph G with \(\chi (G)\) \( = 6\). Further, we show that for every (\(P_6\), diamond)-free graph G, the chromatic number of G is upper bounded by a linear function of its clique number. This generalizes some known results in the literature.

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Acknowledgements

The first author thanks Mathew C.Francis for his comments. The authors are grateful to the anonymous referees for their helpful suggestions and remarks.

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Authors and Affiliations

  1. Computer Science Unit, Indian Statistical Institute, Chennai Centre, Chennai, 600029, India

    T. Karthick

  2. Department of Mathematics, Indian Institute of Technology Madras, Chennai, 600036, India

    Suchismita Mishra

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  1. T. Karthick
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  2. Suchismita Mishra
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Correspondence to T. Karthick.

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Karthick, T., Mishra, S. On the Chromatic Number of (\(P_6\), Diamond)-Free Graphs. Graphs and Combinatorics 34, 677–692 (2018). https://doi.org/10.1007/s00373-018-1905-9

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  • DOI: https://doi.org/10.1007/s00373-018-1905-9

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