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

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 $G$, $\chi''(G)\leq \Delta(G)+2$ if $\Delta(G)\geq 7$. Moreover, $\chi"(G)=\Delta(G)+1$ if $\Delta(G)\geq 10$.