Every planar graph with Δ≥ 8 is totally (Δ+2)-choosable
share
Total coloring is a variant of edge coloring where both vertices and edges are to be colored. A graph is totally k-choosable if for any list assignment of k colors to each vertex and each edge, we can extract a proper total coloring. In this setting, a graph of maximum degree Δ needs at least Δ+1 colors. In the planar case, Borodin proved in 1989 that Δ+2 colors suffice when Δ is at least 9. We show that this bound also holds when Δ is 8.
READ FULL TEXT