List-Recoloring of Sparse Graphs

TL;DR

This paper proves that for certain classes of sparse graphs and list assignments, it is possible to transform one proper coloring into another with a bounded number of recolorings per vertex, confirming existing conjectures.

Contribution

It establishes bounds on recoloring sequences for triangle-free planar graphs and graphs with bounded maximum average degree, confirming conjectures by Dvořák and Feghali.

Findings

Recoloring sequences exist with bounded steps for triangle-free planar graphs.

Recoloring sequences exist with bounded steps for graphs with mad < 17/5 and 6-assignments.

Recoloring sequences exist with bounded steps for graphs with mad < 22/9 and 4-assignments.

Abstract

Fix a graph $G$, a list-assignment $L$ for $G$, and $L$-colorings $\alpha$ and $\beta$. An $L$-recoloring sequence, starting from $\alpha$, recolors a single vertex at each step, so that each resulting intermediate coloring is a proper $L$-coloring. An $L$-recoloring sequence transforms $\alpha$ to $\beta$ if its initial coloring is $\alpha$ and its final coloring is $\beta$. We prove there exists an $L$-recoloring sequence that transforms $\alpha$ to $\beta$ and recolors each vertex at most a constant number of times if (i) $G$ is triangle-free and planar and $L$ is a 7-assignment, or (ii) $\mathrm{mad}(G)<17/5$ and $L$ is a 6-assignment or (iii) $\mathrm{mad}(G)<22/9$ and $L$ is a 4-assignment. Parts (i) and (ii) confirm conjectures of Dvo\v{r}\'{a}k and Feghali.