Total colorings of planar graphs with sparse triangles

Abstract

The total chromatic number of a graph G, denoted by χ″(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 Δ⩾9, then χ″(G)=Δ+1. In this paper, we prove that if G is a planar graph with maximum degree 8, and for every vertex v, v is incident with at most d(v)−2⌊d(v)5⌋ triangles, then χ″(G)=9.