Total coloring of planar graphs without short cycles

Abstract

The total chromatic number of a graph $G$, denoted by $\chi ''(G)$, is the minimum number of colors needed to color the vertices and edges of $G$ such that no two adjacent or incident elements get the same color. It is known that if a planar graph $G$ has maximum degree $\Delta (G)\ge 9$, then $\chi ''(G)=\Delta (G)+1$. In this paper, it is proved that if $G$ is a planar graph with $\Delta (G)\ge 7$, and for each vertex $v$, there is an integer $k_v\in \{3,4,5,6,7,8\}$ such that there is no $k_v$-cycle which contains $v$, then $\chi ''(G)=\Delta (G)+1$.