Paths between colourings of sparse graphs

Abstract

The reconfiguration graph $R_k(G)$ of the $k$-colourings of a graph~$G$ has as vertex set the set of all possible $k$-colourings of $G$ and two colourings are adjacent if they differ on exactly one vertex. We give a short proof of the following theorem of Bousquet and Perarnau (\emph{European Journal of Combinatorics}, 2016). Let $d$ and $k$ be positive integers, $k\ge d+1$. For every $\epsilon >0$ and every graph $G$ with $n$ vertices and maximum average degree $d-\epsilon$, there exists a constant $c=c(d,\epsilon )$ such that $R_k(G)$ has diameter $O(n^c)$. Our proof can be transformed into a simple polynomial time algorithm that finds a path between a given pair of colourings in $R_k(G)$.

3 pages