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Acyclic Coloring of Graphs with Maximum Degree 7

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Abstract

The acyclic chromatic number a(G) of a graph G is the minimum number of colors such that G has a proper vertex coloring and no bichromatic cycles. For a graph G with maximum degree \(\Delta \), Grünbaum (1973) conjectured \(a(G)\le \Delta +1\). Up to now, the conjecture has only been shown for \(\Delta \le 4\). In this paper, it is proved that \(a(G)\le 12\) for \(\Delta =7\), thus improving the result \(a(G)\le 17\) of Dieng et al. (in: Proc. European conference on combinatorics, graph theory and applications, 2010).

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Acknowledgements

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

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Authors and Affiliations

  1. School of Management, Qufu Normal University, Rizhao, 276826, China

    Juan Wang

  2. School of Mathematics, China University of Mining and Technology, Xuzhou, 221116, China

    Juan Wang & Lianying Miao

  3. Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung, 804, Taiwan

    Juan Wang

  4. School of Mathematics and Statistics, Zaozhuang University, Zaozhuang, 277160, China

    Wenyao Song

  5. School of Engineering, Qufu Normal University, Rizhao, 276826, China

    Yunlong Liu

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  1. Juan Wang
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  2. Lianying Miao
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Correspondence to Juan Wang.

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This work was partially supported by the National Natural Science Foundation of China (nos. 11771443,12001481 and 12071265) and Shandong Province Natural Science Foundation (no. ZR2017QF011)

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Wang, J., Miao, L., Song, W. et al. Acyclic Coloring of Graphs with Maximum Degree 7. Graphs and Combinatorics 37, 455–469 (2021). https://doi.org/10.1007/s00373-020-02254-w

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  • DOI: https://doi.org/10.1007/s00373-020-02254-w

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