Reconfiguration of vertex colouring and forbidden induced subgraphs

Abstract

The reconfiguration graph of the k-colourings, denoted Rk(G), is the graph whose vertices are the k-colourings of G and two colourings are adjacent in Rk(G) if they differ in colour on exactly one vertex. In this paper, we investigate the connectivity and diameter of Rk+1(G) for a k-colourable graph G restricted by forbidden induced subgraphs. We show that Rk+1(G) is connected for every k-colourable H-free graph G if and only if H is an induced subgraph of P4 or P3+P1. We also start an investigation into this problem for classes of graphs defined by two forbidden induced subgraphs. We show that if G is a k-colourable (2K2, C4)-free graph, then Rk+1(G) is connected with diameter at most 4n. Furthermore, we show that Rk+1(G) is connected for every k-colourable (P5, C4)-free graph G.