Mixing 3-Colourings in Bipartite Graphs

For a 3-colourable graph G, the 3-colour graph of G, denoted C3(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 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) 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.

• Type: Conference
• Year: 2007
• Where: WG
• Authors: Luis Cereceda, Jan van den Heuvel, Matthew Johnson
• Keywords: 3-colour Graph, 3-colourable Graph G, Bipartite Graphs, Computer Science, WG 2007