# Reconfiguration Graph for Vertex Colourings of Weakly Chordal Graphs

## TL;DR

This paper investigates the structure and properties of reconfiguration graphs for vertex colourings in weakly chordal graphs, revealing disconnectedness in certain cases and polynomial diameter bounds in a new subclass called compact graphs.

## Contribution

It introduces the class of $k$-colourable compact graphs and analyzes their reconfiguration graph diameters, extending understanding of colourings in weakly chordal graphs.

## Key findings

- Existence of $k$-colourable weakly chordal graphs with disconnected reconfiguration graphs for $k+1$ colours.
- Polynomial diameter bound ($O(n^2)$) for reconfiguration graphs of $k$-colourable compact graphs.
- Contains all $k$-colourable co-chordal graphs and specific $(P_5, ar{P}_5, C_5)$-free graphs for $k=3$.

## Abstract

The reconfiguration graph $R_k(G)$ of the $k$-colourings of a graph $G$ contains as its vertex set the $k$-colourings of $G$ and two colourings are joined by an edge if they differ in colour on just one vertex of $G$. We show that for each $k \geq 3$ there is a $k$-colourable weakly chordal graph $G$ such that $R_{k+1}(G)$ is disconnected. We also introduce a subclass of $k$-colourable weakly chordal graphs which we call $k$-colourable compact graphs and show that for each $k$-colourable compact graph $G$ on $n$ vertices, $R_{k+1}(G)$ has diameter $O(n^2)$. We show that this class contains all $k$-colourable co-chordal graphs and when $k = 3$ all $3$-colourable $(P_5, \overline{P_5}, C_5)$-free graphs. We also mention some open problems.