Mixing 3-colourings in bipartite graphs

For a 3-colourable graph GG, the 3-colour graph of GG, denoted C3(G)C3(G), is the graph with node set the proper vertex 3-colourings of GG, and two nodes adjacent whenever the corresponding colourings differ on precisely one vertex of GG. We consider the following question: given GG, how easily can one decide whether or not C3(G)C3(G) is connected? We show that the 3-colour graph of a 3-chromatic graph is never connected, and characterise the bipartite graphs for which C3(G)C3(G) is connected. We also show that the problem of deciding the connectedness of the 3-colour graph of a bipartite graph is coNP-complete, but that restricted to planar bipartite graphs, the question is answerable in polynomial time.