Every planar graph with $\Delta\geqslant 8$ is totally $(\Delta+2)$-choosable

Abstract

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 $\Delta$ needs at least $\Delta+1$ colors. In the planar case, Borodin proved in 1989 that $\Delta+2$ colors suffice when $\Delta$ is at least 9. We show that this bound also holds when $\Delta$ is $8$.