Abstract
A plane graph G is entirely k-colorable if \(V(G)\cup E(G) \cup F(G)\) can be colored with k colors such that any two adjacent or incident elements receive different colors. In 2011, Wang and Zhu conjectured that every plane graph G with maximum degree \(\Delta \ge 3\) and \(G\ne K_4\) is entirely \((\Delta +3)\)-colorable. It is known that the conjecture holds for the case \(\Delta \ge 8\). The condition \(\Delta \ge 8\) is improved to \(\Delta \ge 7\) in this paper.
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Acknowledgements
The authors would like to thank the anonymous referees for their valuable comments and suggestions. The first author was supported partially by NSFC (Nos. 11701541, 11701543) and ZJNSF (No. LQ17A010005), the second author was supported partially by NSFC (Nos. 11801512, 11571315, 11901525), and the third author was supported partially by NSFC (No. 11671053).
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J. Kong: Research supported by NSFC (No. 11701541, 11701543) and ZJNSF (No. LQ17A010005)
X. Hu: Research supported by NSFC (Nos. 11801512, 11571315, 11901525)
Y. Wang: Research supported by NSFC (No. 11671053).
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Kong, J., Hu, X. & Wang, Y. Plane graphs with \(\Delta =7\) are entirely 10-colorable. J Comb Optim 40, 1–20 (2020). https://doi.org/10.1007/s10878-020-00561-9
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DOI: https://doi.org/10.1007/s10878-020-00561-9