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On the diameter of reconfiguration graphs for vertex colourings

Marthe Bonamy 1, * Matthew Johnson 2 Ioannis Lignos 2 Viresh Patel 2 Daniël Paulusma 2
* Corresponding author
1 ALGCO - Algorithmes, Graphes et Combinatoire
LIRMM - Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier
Abstract : The reconfiguration graph of the k-colourings of a graph G contains as its vertex set the proper vertex k-colourings of G, and two colourings are joined by an edge in the reconfiguration graph if they differ in colour on just one vertex of G. We prove that for a graph G on n vertices that is chordal or chordal bipartite, if G is k-colourable, then the reconfiguration graph of its l-colourings, for l>=k+1, is connected and has diameter O(n^2). We show that this bound is asymptotically tight up to a constant factor.
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Marthe Bonamy, Matthew Johnson, Ioannis Lignos, Viresh Patel, Daniël Paulusma. On the diameter of reconfiguration graphs for vertex colourings. EuroComb'11: European Conference on Combinatorics, Graph Theory and Applications, Aug 2011, Budapest, Hungary. pp.161-166. ⟨lirmm-00782875⟩

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