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Reconfiguration graphs for vertex colourings of chordal and chordal bipartite graphs

Author

Listed:
  • Marthe Bonamy

    (Laboratoire d’Informatique, de Robotique et de Microélectronique de Montpellier)

  • Matthew Johnson

    (Durham University)

  • Ioannis Lignos

    (Durham University)

  • Viresh Patel

    (Durham University)

  • Daniël Paulusma

    (Durham University)

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 ℓ-colourings of G is connected and has diameter O(|V|2), for all ℓ≥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).

Suggested Citation

  • Marthe Bonamy & Matthew Johnson & Ioannis Lignos & Viresh Patel & Daniël Paulusma, 2014. "Reconfiguration graphs for vertex colourings of chordal and chordal bipartite graphs," Journal of Combinatorial Optimization, Springer, vol. 27(1), pages 132-143, January.
    • Handle: RePEc:spr:jcomop:v:27:y:2014:i:1:d:10.1007_s10878-012-9490-y
    DOI: 10.1007/s10878-012-9490-y
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    Cited by:

    1. Takehiro Ito & Naonori Kakimura & Naoyuki Kamiyama & Yusuke Kobayashi & Yoshio Okamoto, 2019. "Reconfiguration of maximum-weight b-matchings in a graph," Journal of Combinatorial Optimization, Springer, vol. 37(2), pages 454-464, February.

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