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.KeywordsBipartite GraphHeight FunctionConsecutive VertexPlanar EmbeddingBipartite Planar GraphThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.