A note on orbit graphs of finite groups and colour-clique graphs of Cayley graphs

Let G be a (finite) group and let S be a non-empty subset of G. The vertex set of the orbit graph O(G,S) is the union, over all s ∈ S, of orbits of left translations induced by s. If u and v are distinct vertices (each representing an orbit of some s and t from S), then for any g ∈ G appearing in both orbits there is an edge coloured g in O(G,S) joining u and v. Orbit graphs are an important special case of “G-graphs” introduced by Bretto and Faisant in Math. Slovaca 55 (2005). In this paper we establish a correspondence between simple orbit graphs O(G,S) such that 1 / ∈ S and colour-clique graphs of certain Cayley graphs. This correspondence is used to answer questions about orbit graphs in terms of Cayley graphs.