List-recoloring of two classes of planar graphs

TL;DR

This paper proves that for certain classes of planar graphs and graphs with bounded average degree, there exists an efficient recoloring sequence between two list colorings, with each vertex recolored only a constant number of times.

Contribution

It extends previous theorems by establishing bounded recoloring sequences for specific classes of planar graphs and graphs with bounded average degree.

Findings

Recoloring sequences exist with each vertex recolored a constant number of times.

Results apply to planar graphs with no 3-cycles or intersecting 4-cycles, and to graphs with mad(G) < 5/2.

Strengthens earlier theorems by Cranston.

Abstract

For a graph $G$ with a list assignment $L$ and two $L$-colorings $\alpha$ and $\beta$, an $L$-recoloring sequence from $\alpha$ to $\beta$ is a sequence of proper $L$-colorings where consecutive colorings differ at exactly one vertex. We prove the existence of such a recoloring sequence in which every vertex is recolored at most a constant number of times under two conditions: (i) $G$ is planar, contains no $3$-cycles or intersecting $4$-cycles, and $L$ is a $6$-assignment; or (ii) the maximum average degree of $G$ satisfies $\mathrm{mad}(G) < \frac{5}{2}$ and $L$ is a $4$-assignment. These results strengthen two theorems previously established by Cranston.