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Finding paths between 3-colorings

By Luis Cereceda, Jan van den Heuvel and Matthew Johnson

Abstract

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(vertical bar V(G)vertical bar(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(vertical bar V(G)vertical bar(2)) recoloring steps

Topics: QA Mathematics
Publisher: John Wiley & Sons
Year: 2011
DOI identifier: 10.1002/jgt.20514
OAI identifier: oai:eprints.lse.ac.uk:36005
Provided by: LSE Research Online
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