Finding paths between 3-colorings. (English) Zbl 1216.05154
Summary: Given a 3-colorable graph \(G\) together with two proper vertex 3-colorings \(\alpha\) and \(\beta\) of \(G\), consider the following question: is it possible to transform \(\alpha\) into \(\beta\) by recoloring vertices of \(G\) one at a time, making sure that all intermediate colorings are proper 3-colorings? We prove that this question is answerable in polynomial time. We do so by characterizing the instances \(G\), \(\alpha\), \(\beta\), where the transformation is possible; the proof of this characterization is via an algorithm that either finds a sequence of recolorings between \(\alpha\) and \(\beta\), or exhibits a structure which proves that no such sequence exists.
In the case that a sequence of recolorings does exist, the algorithm uses \(O(|V(G)|^2)\) recoloring steps and in many cases returns a shortest sequence of recolorings. We also exhibit a class of instances \(G\), \(\alpha\), \(\beta\) that require \(\Omega(|V(G)|^2)\) recoloring steps.
In the case that a sequence of recolorings does exist, the algorithm uses \(O(|V(G)|^2)\) recoloring steps and in many cases returns a shortest sequence of recolorings. We also exhibit a class of instances \(G\), \(\alpha\), \(\beta\) that require \(\Omega(|V(G)|^2)\) recoloring steps.
References:
| [1] | J. Billingham R. Leese H. Rajaniemi http://www.smithinst.ac.uk/Projects/ESGI53/ESGI53-Motorola/Report/ |
| [2] | Bonsma, Finding paths between graph colourings: PSPACE-completeness and superpolynomial distances, Theoret Comput Sci 410 pp 5215– (2009) · Zbl 1177.05112 · doi:10.1016/j.tcs.2009.08.023 |
| [3] | Cereceda, Mixing 3-colourings in bipartite graphs, Eur J Combin 30 pp 1593– (2009) · Zbl 1198.05040 · doi:10.1016/j.ejc.2009.03.011 |
| [4] | Cereceda, Connectedness of the graph of vertex-colourings, Discrete Math 308 pp 913– (2008) · Zbl 1133.05053 · doi:10.1016/j.disc.2007.07.028 |
| [5] | Diestel, Graph Theory (2005) |
| [6] | Jerrum, A very simple algorithm for estimating the number of k-colorings of a low degree graph, Random Struct Algor 7 pp 157– (1995) · Zbl 0833.60070 · doi:10.1002/rsa.3240070205 |
| [7] | Jerrum, Counting, Sampling and Integrating: Algorithms and Complexity (2003) · doi:10.1007/978-3-0348-8005-3 |
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