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