Reconfiguration Graph for Vertex Colourings of Weakly Chordal Graphs
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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 answer a question of Bonamy, Johnson, Lignos, Patel and Paulusma by constructing for each k ≥ 3 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-colour compact and show that for each k-colour 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, P_5, C_5)-free graphs. We also mention some open problems.
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