Connectedness of the graph of vertex-colourings Original Research Article

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