Optimal List Recoloring of Subcubic Graphs and Complete Multipartite Graphs

TL;DR

This paper proves a conjecture about the diameter of list recoloring graphs for two classes of graphs, advancing understanding of graph reconfiguration complexity.

Contribution

It confirms the conjecture for subcubic graphs and complete multipartite graphs, solving two open problems in graph recoloring theory.

Findings

Confirmed the conjecture for subcubic graphs.

Confirmed the conjecture for complete multipartite graphs.

Advances understanding of reconfiguration graph diameters.

Abstract

For a list-assignment $L$, the reconfiguration graph $C_L(G)$ of a graph $G$ is the graph whose vertices are proper $L$-colorings of $G$ and whose edges link two colorings that differ on only one vertex. If $|L(v)| \ge d(v) + 2$ for every vertex of $G$, it is known that $C_L(G)$ is connected. In this case, Cambie et al. investigated the diameter of $C_L(G)$. They conjectured that $diam(C_L(G)) \le n(G) + \mu(G)$ with $\mu(G)$ the size of a maximum matching of $G$ and proved several results towards this conjecture. We answer to two of their open problems by proving the conjecture for two classes of graphs, namely subcubic graphs and complete multipartite graphs.