Connectedness of Strong k-Colour Graphs

Abstract

For a positive integer k and a graph G, we consider proper vertex-colourings of G with k colours in which all k colours are actually used. We call such a colouring a strong k-colouring. The strong k-colour graph of G, S_k(G), is the graph that has all the strong k-colourings of G as its vertex set, and two colourings are adjacent in S_k(G) if they differ in colour on only one vertex of G. In this paper, we show some results related to the question : For what G and k is S_k(G) connected ?