Recolouring weakly chordal graphs and the complement of triangle-free graphs
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For a graph G, the k-recolouring graph 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 ≥ 1, there exists a k-colourable weakly chordal graph G where 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, ℛ_k+1(G) is connected with diameter at most 4|V(G)|.
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