10-list Recoloring of Planar Graphs
Daniel W. Cranston
TL;DR
This work proves that planar graphs with a fixed list assignment of size $|L(v)|=10$ admit $k$-good recolorings between any two colorings, with recolorings bounded linearly in the number of vertices. The authors introduce an Extension Lemma, a 'good subgraph' inductive framework, and a novel deferral/out-tree technique to certify reducible configurations without computer verification, enabling a linear bound on recolor length. A discharging argument with 36 reducible configurations (later refined to 520) eliminates the possibility of a minimal counterexample, confirming a conjecture by Dvorak and Feghali for planar graphs. The methods offer a potentially broad toolkit for tackling reconfiguration problems via local reducibility plus global discharging, and may extend to other coloring or correspondence-coloring settings.
Abstract
Fix a planar graph $G$ and a list-assignment $L$ with $|L(v)|=10$ for all $v\in V(G)$. Let $α$ and $β$ be $L$-colorings of $G$. A recoloring sequence from $α$ to $β$ is a sequence of $L$-colorings, beginning with $α$ and ending with $β$, such that each successive pair in the sequence differs in the color on a single vertex of $G$. We show that there exists a constant $C$ such that for all choices of $α$ and $β$ there exists a recoloring sequence $σ$ from $α$ to $β$ that recolors each vertex at most $C$ times. In particular, $σ$ has length at most $C|V(G)|$. This confirms a conjecture of Dvořák and Feghali. For our proof, we introduce a new technique for quickly showing that many configurations are reducible. We believe this method may be of independent interest and will have application to other problems in this area.