Centre for Discrete and Applicable Mathematics 

CDAM Research Report, LSECDAM200706February 2007 
Mixing 3Colourings in Bipartite Graphs
Luis Cereceda, Jan van den Heuvel, and Matthew Johnson
For a 3colourable graph G, the 3colour graph of G, denoted C_{3}(G), is the graph with node set the proper vertex 3colourings 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 C_{3}(G) is connected? We show that the 3colour graph of a 3chromatic 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 3colour graph of a bipartite graph is coNPcomplete, but that restricted to planar bipartite graphs, the question is answerable in polynomial time.A PDF file (258 kB) with the full contents of this report can be downloaded by clicking here.
(This paper supersedes the earlier paper Mixing 3colourings in Bipartite Graphs, which is obsolete.)
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