Abstract
A graph is \((d_1,\ldots ,d_k)\)-colorable if the vertex set can be partitioned into k sets \(V_{1},\ldots ,V_{k}\) where the maximum degree of the graph induced by \(V_{i}\) is at most \(d_{i}\) for each i, where \(1\le i\le k\). In this paper, we prove that every planar graph without 4-cycles and 5-cycles is (3,3)-colorable, which improves the result of Sittitrai and Nakprasit, who proved that every planar graph without 4-cycles and 5-cycles is (3, 5)-colorable (Sittitrai and Nakprasit in Discrete Math 341:2142–2150, 2018).
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The authors would like to thank the referees for the valuable comments and suggestions which are helpful for improvement of this representation.
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This article was funded by Natural Science Foundation of China (Grant no. 12031018).
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X. Li: Supported by National Natural Science Foundation of China (12031018). J.-B. Lv: Supported by the Science and technology project of Guanxi (Guike AD 21220114).
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Li, X., Liu, J. & Lv, JB. Every planar graph without 4-cycles and 5-cycles is (3,3)-colorable. Graphs and Combinatorics 39, 118 (2023). https://doi.org/10.1007/s00373-023-02713-0
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DOI: https://doi.org/10.1007/s00373-023-02713-0