Centre for Discrete and Applicable Mathematics

 CDAM Research Report, LSE-CDAM-2005-11

July 2005

Connectedness of the Graph of Vertex-Colourings

Luis Cereceda, Jan van den Heuvel and Matthew Johnson

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 connectivity of Ck(G) are proved. In particular it is shown that if G has chromatic number k 2 {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.

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