Colouring clique-hypergraphs of circulant graphs 1

A clique-colouring of a graph G is a colouring of the vertices of G so that no maximal clique of size at least two is monochromatic. The clique-hypergraph, H(G), of a graph G has V(G) as its set of vertices and the maximal cliques of G as its hyperedges. A vertex-colouring of H(G) is a clique-colouring of G. Determining the clique-chromatic number, the least number for which a graph G admits a clique-colouring, is known to be NP-hard. We establish that the clique-chromatic number for powers of cycles is equal to two, except for odd cycles of size at least five, that need three colours. For odd-seq circulant graphs, we show that their clique-chromatic number is at most four, and determine the cases when it is equal to two.