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

TL;DR

This paper proves that every planar graph with maximum degree at least 8 is totally $( ext{max degree}+2)$-choosable, extending Borodin's result from degree 9 to degree 8.

Contribution

It extends the known bound for total choosability of planar graphs from degree 9 to degree 8.

Findings

Planar graphs with $ ext{max degree} \, \geq 8$ are totally $(\text{max degree}+2)$-choosable.

The bound $ ext{max degree}+2$ is sufficient for total coloring in these graphs.

The result completes the understanding of total choosability for planar graphs with high maximum degree.

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 $8$.