Reconfiguration graphs for vertex colourings of chordal and chordal bipartite graphs

  • Authors:

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

  • Affiliations:

  • Laboratoire d'Informatique, de Robotique et de Microélectronique de Montpellier, Montpellier, France;School of Engineering and Computing Sciences, Durham University, Durham, UK;School of Engineering and Computing Sciences, Durham University, Durham, UK;School of Engineering and Computing Sciences, Durham University, Durham, UK;School of Engineering and Computing Sciences, Durham University, Durham, UK

  • Venue:
  • Journal of Combinatorial Optimization
  • Year:
  • 2014

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Abstract

A k-colouring of a graph G=(V,E) is a mapping c:V驴{1,2,驴,k} such that c(u)驴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 ℓ-colourings of G is connected and has diameter O(|V|2), for all ℓ驴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驴2 there is a k-colourable chordal graph G whose reconfiguration graph of the (k+1)-colourings has diameter 驴(|V|2).