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Connectedness of the graph of vertex-colourings

Cereceda, Luis; van den Heuvel, Jan; Johnson, Matthew

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Luis Cereceda

Jan van den Heuvel


For a positive integer k and a graph G, the k-colour graph of G , Ck(G), is the graph that has the proper k-vertex-colourings of G as its vertex set, and two k -colourings are joined by an edge in Ck(G) if they differ in colour on just one vertex of G . In this note some results on the connectedness of Ck(G) are proved. In particular, it is shown that if G has chromatic number k∈{2,3}, then Ck(G) is not connected. On the other hand, for k⩾4 there are graphs with chromatic number k for which Ck(G) is not connected, and there are k -chromatic graphs for which Ck(G) is connected.


Cereceda, L., van den Heuvel, J., & Johnson, M. (2008). Connectedness of the graph of vertex-colourings. Discrete Mathematics, 308(5-6), 913-919.

Journal Article Type Article
Publication Date Mar 28, 2008
Deposit Date Oct 7, 2008
Publicly Available Date Dec 11, 2015
Journal Discrete mathematics.
Print ISSN 0012-365X
Publisher Elsevier
Peer Reviewed Peer Reviewed
Volume 308
Issue 5-6
Pages 913-919
Keywords Vertex colouring, k-colour graph, Glauber dynamics.


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