Connectedness of the graph of vertex-colourings

For a positive integer k and a graph G, the k-colour graph of G, View the MathML source, is the graph that has the proper k-vertex-colourings of G as its vertex set, and two k-colourings are joined by an edge in View the MathML source if they differ in colour on just one vertex of G. In this note some results on the connectedness of View the MathML source are proved. In particular, it is shown that if G has chromatic number kset membership, variant{2,3}, then View the MathML source is not connected. On the other hand, for kgreater-or-equal, slanted4 there are graphs with chromatic number k for which View the MathML source is not connected, and there are k-chromatic graphs for which View the MathML source is connected.