Total Colorings of $F_5$-free Planar Graphs with Maximum Degree 8

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 ∆ > 9, then χ′′(G) = ∆ + 1. The join K1 ∨ Pn of K1 and Pn is called a fan graph Fn. In this paper, we prove that if G is an F5-free planar graph with maximum degree 8, then χ′′(G) = 9.