Connectedness of the graph of vertex-colourings

For a positive integer k and a graph G, the k-colour graph of G  , Ck(G)Ck(G), 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 Ck(G)Ck(G) if they differ in colour on just one vertex of G  . In this note some results on the connectedness of Ck(G)Ck(G) are proved. In particular, it is shown that if G   has chromatic number k∈{2,3}k∈{2,3}, then Ck(G)Ck(G) is not connected. On the other hand, for k⩾4k⩾4 there are graphs with chromatic number k   for which Ck(G)Ck(G) is not connected, and there are k  -chromatic graphs for which Ck(G)Ck(G) is connected.