List-recoloring of two classes of planar graphs

TL;DR

The paper studies reconfiguration between proper $L$-colorings of graphs, seeking recoloring sequences where each vertex is recolored a bounded number of times. It proves two main results using a unified discharging-based method anchored by a Key Lemma: (i) for planar graphs with no $3$-cycles or intersecting $4$-cycles and a $6$-assignment $L$, there exists an $L$-coloring sequence from any given pair of colorings in which every vertex is recolored at most $48$ times; (ii) for graphs with $\mathrm{mad}(G)<\frac{5}{2}$ and a $4$-assignment $L$, there is an $L$-coloring sequence with each vertex recolored at most $18$ times (in particular for planar graphs of girth at least $10$). The approach hinges on the Key Lemma for extending recolorings and a careful discharging analysis to rule out unavoidable configurations, thereby bounding the recoloring complexity and extending Cranston’s prior results on recoloring diameters of sparse planar graphs.

Abstract

For a graph $G$ with a list assignment $L$ and two $L$-colorings $α$ and $β$, an $L$-recoloring sequence from $α$ to $β$ 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.

Theorems & Definitions (30)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1: Key Lemma C22
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• proof