The entire chromatic number of graphs embedded on the torus with large maximum degree

An embedded graph G=(V,E,F) on the torus is entirely k-colorable if V∪E∪F can be colored with k colors such that any two adjacent or incident elements receive different colors. In this paper, we prove that every embedded graph G on the torus with maximum degree Δ≥10 is entirely (Δ+2)-colorable.