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Entire Coloring of Graphs Embedded in a Surface of Nonnegative Characteristic

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Abstract

Let G be a graph embedded in a surface of nonnegative characteristic with maximum degree \(\varDelta \). The entire chromatic number \(\chi _\mathrm{vef}\) of G is the least number of colors such that any two adjacent or incident elements in \(V(G)\cup E(G) \cup F(G)\) have different colors. In this paper, we prove that \(\chi _\mathrm{vef}(G)\le \varDelta +4\) if \(\varDelta \ge 6\), and \(\chi _\mathrm{vef}(G)\le \varDelta +5\) if \(\varDelta \le 5\).

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Acknowledgements

The authors are most grateful to the referees whose comments led to an improved version of our paper. The first author’s research is supported by NSFC (nos. 11801512, 11701541 and 11571315). The second author’s research is supported by NSFC (no. 11771402). The third author’s research is supported by NSFC (no. 11671053).

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Authors and Affiliations

  1. School of Science, Zhejiang University of Science and Technology, Hangzhou, 310023, China

    Xiaoxue Hu

  2. Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, China

    Weifan Wang

  3. School of Management, Beijing University of Chinese Medicine, Beijing, 100029, China

    Yiqiao Wang

  4. Department of Mathematics, Statistics and Computer Science, St. Francis Xavier University, Antigonish, Nova Scotia, Canada

    Ping Wang

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  1. Xiaoxue Hu
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  2. Weifan Wang
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  3. Yiqiao Wang
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  4. Ping Wang
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Correspondence to Weifan Wang.

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Hu, X., Wang, W., Wang, Y. et al. Entire Coloring of Graphs Embedded in a Surface of Nonnegative Characteristic. Graphs and Combinatorics 34, 1489–1506 (2018). https://doi.org/10.1007/s00373-018-1971-z

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  • DOI: https://doi.org/10.1007/s00373-018-1971-z

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