Total chromatic number of graphs with small genus

Abstract

Abstract Given a graph G, a total k-coloring of G is a simultaneous coloring of the vertices and edges of G with k colors. Denote χve (G) the total chromatic number of G, and c(Σ) the Euler characteristic of a surfase Σ. In this paper, we prove that for any simple graph G which can be embedded in a surface Σ with Euler characteristic c(Σ), χve (G) = Δ (G) + 1 if c(Σ) > 0 and Δ (G) ≥ 13, or, if c(Σ) = 0 and Δ (G) ≥ 14. This result generalizes results in [3], [4], [5] by Borodin.