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Coloring Squares of Planar Graphs with Maximum Degree at Most Five

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Abstract

The square \(G^2\) of a graph G is the graph on the same vertex set as G in which two vertices u, v are adjacent if and only if \(uv\in E(G)\), or u and v have at least one common neighbor in G. We denote by \(\chi _2(G)\) the chromatic number of \(G^2\), which is the least integer k such that \(G^2\) admits a proper k-coloring. In this paper, we show that if G is a planar graph with maximum degree at most 5, then \(\chi _2(G)\le 18\), which improves a result given by Chen et al. (Discrete Math 345:112766, 2022).

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Acknowledgements

We would like to thank the referees for providing some very helpful suggestions for revising this paper.

Funding

This work was supported by National Natural Science Foundation of China (Grant numbers 12071077, 11771443 and 12071265). The first author was supported by National Natural Science Foundation of China (Grant No. 12071077). The third author was supported by National Natural Science Foundation of China (Grant No. 11771443, 12071265).

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Authors and Affiliations

  1. Center for Discrete Mathematics, Fuzhou University, Fujian, 350003, China

    Jianfeng Hou & Yindong Jin

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

    Lianying Miao & Qian Zhao

Authors
  1. Jianfeng Hou
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  2. Yindong Jin
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  3. Lianying Miao
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  4. Qian Zhao
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Contributions

The authors make their contributions equally. All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by JH, YiJ, LM and QZ. The first draft of the manuscript was written by JH, YJ, LM and QZ. And all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

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Correspondence to Yindong Jin.

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We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted. The authors have no relevant financial or non-financial interests to disclose.

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Jianfeng Hou: Supported by the NSFC (no. 12071077).

Lianying Miao: Supported by the NSFC (nos. 11771443, 12071265).

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Hou, J., Jin, Y., Miao, L. et al. Coloring Squares of Planar Graphs with Maximum Degree at Most Five. Graphs and Combinatorics 39, 20 (2023). https://doi.org/10.1007/s00373-023-02615-1

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  • DOI: https://doi.org/10.1007/s00373-023-02615-1

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