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arXiv · 2406.02792

Weak Degeneracy of Planar Graphs

Anton Bernshteyn·Eugene Lee·Evelyne Smith-Roberge
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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.

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Anton Bernshteyn, Eugene Lee, Evelyne Smith-Roberge. 2025-06-04. Weak Degeneracy of Planar Graphs. https://arxiv.org/abs/2406.02792

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