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.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
Keywords
These 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.
References
Baxter, R.J.: Exactly Solved Models in Statistical Mechanics. Academic Press, New York (1982)
Cereceda, L., van den Heuvel, J., Johnson, M.: Connectedness of the graph of vertex-colourings. Discrete Math. (to appear)
Diestel, R.: Graph Theory, 2nd edn. Springer, Heidelberg (2000)
Goldberg, L.A., Martin, R., Paterson, M.: Random sampling of 3-colorings in ℤ2. Random Structures Algorithms 24, 279–302 (2004)
Jerrum, M.: A very simple algorithm for estimating the number of k-colourings of a low degree graph. Random Structures Algorithms 7, 157–165 (1995)
Jerrum, M.: Counting, Sampling and Integrating: Algorithms and Complexity. Birkhäuser Verlag, Basel (2003)
Vikas, N.: Computational complexity of compaction to irreflexive cycles. J. Comput. Syst. Sci. 68, 473–496 (2004)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Cereceda, L., van den Heuvel, J., Johnson, M. (2007). Mixing 3-Colourings in Bipartite Graphs. In: Brandstädt, A., Kratsch, D., Müller, H. (eds) Graph-Theoretic Concepts in Computer Science. WG 2007. Lecture Notes in Computer Science, vol 4769. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74839-7_17
Download citation
DOI: https://doi.org/10.1007/978-3-540-74839-7_17
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-74838-0
Online ISBN: 978-3-540-74839-7
eBook Packages: Computer ScienceComputer Science (R0)