Journal of Combinatorial Optimization

, Volume 27, Issue 1, pp 132–143 | Cite as

Reconfiguration graphs for vertex colourings of chordal and chordal bipartite graphs

  • Marthe Bonamy
  • Matthew Johnson
  • Ioannis Lignos
  • Viresh Patel
  • Daniël Paulusma
Article

Abstract

A k-colouring of a graph G=(V,E) is a mapping c:V→{1,2,…,k} such that c(u)≠c(v) whenever uv is an edge. The reconfiguration graph of the k-colourings of G contains as its vertex set the k-colourings of G, and two colourings are joined by an edge if they differ in colour on just one vertex of G. We introduce a class of k-colourable graphs, which we call k-colour-dense graphs. We show that for each k-colour-dense graph G, the reconfiguration graph of the -colourings of G is connected and has diameter O(|V|2), for all k+1. We show that this graph class contains the k-colourable chordal graphs and that it contains all chordal bipartite graphs when k=2. Moreover, we prove that for each k≥2 there is a k-colourable chordal graph G whose reconfiguration graph of the (k+1)-colourings has diameter Θ(|V|2).

Keywords

Reconfigurations Graph colouring Graph diameter Chordal graphs 

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Marthe Bonamy
    • 1
  • Matthew Johnson
    • 2
  • Ioannis Lignos
    • 2
  • Viresh Patel
    • 2
  • Daniël Paulusma
    • 2
  1. 1.Laboratoire d’Informatique, de Robotique et de Microélectronique de MontpellierMontpellierFrance
  2. 2.School of Engineering and Computing SciencesDurham UniversityDurhamUK

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