A Thomassen-type method for planar graph recoloring

TL;DR

This paper proves bounds on the diameter of the reconfiguration graph of k-colorings for planar graphs, using a Thomassen-inspired list coloring technique, with improved bounds for triangle-free graphs.

Contribution

Introduces a Thomassen-type list coloring method to analyze the reconfiguration graph diameters of planar graphs.

Findings

For any planar graph with n vertices, R_{10}(G) has diameter at most 8n.

For triangle-free planar graphs, R_7(G) has diameter at most 7n.

Abstract

The reconfiguration graph $R_k(G)$ for the $k$-colorings of a graph $G$ has as vertices all possible $k$-colorings of $G$ and two colorings are adjacent if they differ in the color of exactly one vertex. We use a list coloring technique inspired by results of Thomassen to prove that for a planar graph $G$ with $n$ vertices, $R_{10}(G)$ has diameter at most $8n$, and if $G$ is triangle-free, then $R_7(G)$ has diameter at most $7n$.