The total coloring of $K_5$-minor-free graphs

Abstract

A total $k$-coloring of a graph $G$ is a coloring of $V(G)\cup E(G)$ 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 total $k$-coloring. In the paper, it is proved that for any $K_5$-minor-free graph $G$, ${\chi }^{\prime \prime }(G)\le \Delta (G)+2$ if $\Delta (G)\ge 7$. Moreover, $\chi "(G)=\Delta (G)+1$ if $\Delta (G)\ge 10$.