Weak Degeneracy of Planar Graphs

TL;DR

This paper proves that planar graphs are weakly 4-degenerate, providing a stronger understanding of their coloring properties and correcting a previous mistake, which enhances the classical result that planar graphs are 5-list-colorable.

Contribution

The authors establish that planar graphs are weakly 4-degenerate, correcting earlier errors and strengthening the known coloring bounds for planar graphs.

Findings

Planar graphs are weakly 4-degenerate.

Weak degeneracy bounds imply improved coloring parameters.

Correction of previous proof errors regarding planar graph degeneracy.

Abstract

The weak degeneracy of a graph $G$ is a numerical parameter that was recently introduced by the first two authors with the aim of understanding the power of greedy algorithms for graph coloring. Every $d$-degenerate graph is weakly $d$-degenerate, but the converse is not true in general (for example, all connected $d$-regular graphs except cycles and cliques are weakly $(d-1)$-degenerate). If $G$ is weakly $d$-degenerate, then the list-chromatic number of $G$ is at most $d+1$, and the same upper bound holds for various other parameters such as the DP-chromatic number and the paint number. Here we rectify a mistake in a paper of the first two authors and give a correct proof that planar graphs are weakly $4$-degenerate, strengthening the famous result of Thomassen that planar graphs are $5$-list-colorable.