On the diameter of reconfiguration graphs for vertex colourings

The reconfiguration graph of the k-colourings of a graph G contains as its vertex set the proper vertex k-colourings of G, and two colourings are joined by an edge in the reconfiguration graph if they differ in colour on just one vertex of G. We prove that for a graph G on n vertices that is chordal or chordal bipartite, if G is k-colourable, then the reconfiguration graph of its ℓ-colourings, for ℓ ⩾ k + 1 , is connected and has diameter O ( n 2 ) . We show that this bound is asymptotically tight up to a constant factor.