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