CDAM: Computational, Discrete and Applicable Mathematics@LSE

 CDAM Research Report, LSE-CDAM-2007-31

December 2007


Finding Paths Between 3-Colourings

Luis Cereceda, Jan van den Heuvel, and Matthew Johnson

Given a 3-colourable graph G and two proper vertex 3-colourings α and β of G, consider the following question: is it possible to transform α into β by recolouring vertices of G one at a time, making sure that all intermediate colourings are proper 3-colourings?
We prove that this question is answerable in polynomial time. We do so by characterising the instances G,α,β where the transformation is possible; the proof of this characterisation is via an algorithm that either finds a sequence of recolourings between α and β, or exhibits a structure which proves that no such sequence exists. In the case that a sequence of recolourings does exist, the algorithm uses O(|V(G)|2) recolouring steps and in many cases returns a shortest sequence of recolourings. We also exhibit a class of instances G,α,β that require Ω(|V(G)|2) recolouring steps.

A PDF file (227 kB) with the full contents of this report can be downloaded by clicking here.

Alternatively, if you would like to get a free hard copy of this report, please send the number of this report, LSE-CDAM-2007-31, together with your name and postal address to:
CDAM Research Reports Series
Centre for Discrete and Applicable Mathematics
London School of Economics
Houghton Street
London WC2A 2AE, U.K.
Phone: +44(0)-20-7955 7494.
Fax: +44(0)-20-7955 6877.
Email: info@maths.lse.ac.uk 


Introduction to the CDAM Research Report Series.
CDAM Homepage.


Copyright © London School of Economics & Political Science 2007