Paths between colourings of sparse graphs

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 the colour of exactly one vertex. We give a short proof of the following theorem of Bousquet and Perarnau (European Journal of Combinatorics, 2016). Let d and k be positive integers, k >= d + 1. For every e > O 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). (C) 2018 Elsevier Ltd. All rights reserved.