A sufficient condition for planar graphs with maximum degree eight to be totally 9-colorable

A total coloring of a graph $G$ is a coloring of the vertices and edges such that two adjacent or incident elements receive different colors. The minimum number of colors required for a total coloring of a graph $G$ is called the total chromatic number, denoted by ${\chi }^{\mathrm{\prime }\mathrm{\prime }}(G)$. Let $G$ be a planar graph of maximum degree eight. It is known that $9\le {\chi }^{\mathrm{\prime }\mathrm{\prime }}(G)\le 10$. We here prove that ${\chi }^{\mathrm{\prime }\mathrm{\prime }}(G)=9$ when the graph does not contain any subgraph isomorphic to a $4$-fan.