Recolouring weakly chordal graphs and the complement of triangle-free graphs

TL;DR

This paper investigates the properties of the $k$-recolouring graph for specific classes of graphs, showing that connectivity can fail even for weakly chordal graphs but holds with bounded diameter for certain triangle-free graphs.

Contribution

It proves the existence of weakly chordal graphs with disconnected recolouring graphs and establishes connectivity and diameter bounds for $3K_1$-free graphs.

Findings

Existence of weakly chordal graphs with disconnected $ ext{Recolouring}_k(G)$

Connectivity of $ ext{Recolouring}_{k+1}(G)$ for $3K_1$-free graphs

Diameter of $ ext{Recolouring}_{k+1}(G)$ is at most $4|V(G)|$

Abstract

For a graph $G$, the $k$-recolouring graph $\mathcal{R}_k(G)$ is the graph whose vertices are the $k$-colourings of $G$ and two colourings are joined by an edge if they differ in colour on exactly one vertex. We prove that for all $n \ge 1$, there exists a $k$-colourable weakly chordal graph $G$ where $\mathcal{R}_{k+n}(G)$ is disconnected, answering an open question of Feghali and Fiala. We also show that for every $k$-colourable $3K_1$-free graph $G$, $\mathcal{R}_{k+1}(G)$ is connected with diameter at most $4|V(G)|$.