10-list Recoloring of Planar Graphs

TL;DR

This paper proves that for any two list colorings of a planar graph with lists of size 10, there exists a bounded-length recoloring sequence transforming one into the other, confirming a conjecture and introducing a new reducibility technique.

Contribution

The paper establishes a universal bound on recoloring steps between any two 10-list colorings of planar graphs, confirming a key conjecture and presenting a novel reducibility method.

Findings

Existence of a constant C bounding recoloring steps

Recoloring sequences have length at most C|V(G)|

New technique for showing reducibility of configurations

Abstract

Fix a planar graph $G$ and a list-assignment $L$ with $|L(v)|=10$ for all $v\in V(G)$. Let $\alpha$ and $\beta$ be $L$-colorings of $G$. A recoloring sequence from $\alpha$ to $\beta$ is a sequence of $L$-colorings, beginning with $\alpha$ and ending with $\beta$, 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 $\alpha$ and $\beta$ there exists a recoloring sequence $\sigma$ from $\alpha$ to $\beta$ that recolors each vertex at most $C$ times. In particular, $\sigma$ has length at most $C|V(G)|$. This confirms a conjecture of Dvo\v{r}\'{a}k 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.