Total coloring of planar graphs without short cycles

Abstract

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