Mixing 3-Colourings in Bipartite Graphs

Abstract

For a 3-colourable graph G, the 3-colour graph of G, denoted $\mathcal{C}_3(G)$, 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 we decide whether or not $\mathcal{C}_3(G)$ is connected? We show that the 3-colour graph of a 3-chromatic graph is never connected, and characterise the bipartite graphs for which $\mathcal{C}_3(G)$ 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.