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 Total colorings of planar graphs with maximum degree at least 7 and without adjacent 5-cycles Bull. Korean Math. Soc. 2016 Vol. 53, No. 1, 139-151 https://doi.org/10.4134/BKMS.2016.53.1.139Printed January 31, 2016 Xiang Tan Shandong University of Finance and Economics Abstract : A $k$-total-coloring of a graph $G$ is a coloring of $V\cup E$ using $k$ colors such that no two adjacent or incident elements receive the same color. The total chromatic number $\chi''(G)$ of $G$ is the smallest integer $k$ such that $G$ has a $k$-total-coloring. Let $G$ be a planar graph with maximum degree $\Delta$. In this paper, it's proved that if $\Delta\geq 7$ and $G$ does not contain adjacent 5-cycles, then the total chromatic number $\chi''(G)$ is $\Delta+1$. Keywords : planar graph, total coloring, adjacent 5-cycle MSC numbers : 05C15 Downloads: Full-text PDF