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 $\Delta ≥ 9$, then $\chi′′(G) = \Delta + 1$. The join $K_1 \vee P_n$ of $K_1$ and $P_n$ is called a fan graph $F_n$. In this paper, we prove that if $G$ is a $F_5$-free planar graph with maximum degree 8, then $\chi′′(G) = 9$.
Highlights
All graphs considered in this paper are simple, finite, and undirected
We prove that if G is an F5-free planar graph with maximum degree 8, χ (G) = 9
We follow [2] for the terminology and notation not defined here
Summary
All graphs considered in this paper are simple, finite, and undirected. We follow [2] for the terminology and notation not defined here. According to [9], planar graphs with ∆ 9 have a total (∆ + 1)-coloring, so to prove Theorem 1, in the following we assume that ∆ = 8. Note that every nice coloring can be greedily extended to a total 9-coloring of G, since each 4−-vertex is adjacent to at most four vertices and incident with at most four the electronic journal of combinatorics 21(1) (2014), #P1.56 v v v edges.
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