Mixing 3-colourings in bipartite graphs

For a 3-colourable graph G , the 3-colour graph of G , denoted C 3 ( G ) , is the graph with node set the proper vertex 3-colourings of G , and two nodes adjacent whenever the corresponding colourings differ on precisely one vertex of G . We consider the following question: given G , how easily can one decide whether or not C 3 ( 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 C 3 ( 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.