Colouring Clique-Hypergraphs of Circulant Graphs

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, , 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 is a clique-colouring of G. Determining the clique-chromatic number, the least number of colours for which a graph G admits a clique-colouring, is known to be NP-hard. In this work, we establish that the clique-chromatic number of 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. Similar bounds for the chromatic number of these graphs are also obtained.