Abstract
A \(k\)-total-coloring of a graph \(G\) is a coloring of vertex set and edge set using \(k\) colors such that no two adjacent or incident elements receive the same color. In this paper, we prove that if \(G\) is a planar graph with maximum \(\Delta \ge 8\) and every 6-cycle of \(G\) contains at most one chord or any chordal 6-cycles are not adjacent, then \(G\) has a \((\Delta +1)\)-total-coloring.
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Acknowledgments
The total work is supported by a research Grant NSFC (11271006) of China. Huijuan Wang work is supported by a research Grant NSFC (11201440) of China.
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Communicated by Xueliang Li.
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Xu, R., Wu, J. & Wang, H. Total Coloring of Planar Graphs Without Some Chordal 6-cycles. Bull. Malays. Math. Sci. Soc. 38, 561–569 (2015). https://doi.org/10.1007/s40840-014-0036-6
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DOI: https://doi.org/10.1007/s40840-014-0036-6