Mixing 3-colourings in bipartite graphs

Authors:
Luis Cereceda;Jan van den Heuvel;Matthew Johnson
Affiliations:
Department of Mathematics, London School of Economics, Houghton Street, London WC2A 2AE, UK;Department of Mathematics, London School of Economics, Houghton Street, London WC2A 2AE, UK;Department of Computer Science, Durham University, Science Laboratories, South Road, Durham DH1 3LE, UK
Venue:
European Journal of Combinatorics
Year:
2009

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Abstract

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.