Two-fold orbital digraphs and other constructions

In this paper we shall present a natural generalisation of orbital graphs. If Gamma is a subgroup of Sym(n) × Sym(n), V an n-set, and (u,v)\in V x V, then the orbit of (u,v) under the action of Gamma will be the arc-set of a digraph G with vertex-set V. Such a G will be called a two-fold orbital digraph (TOD). We shall emphasise properties of G which are markedly different from those of orbital graphs, focusing, in particular, on the case when G is disconnected, since this case brings out very sharply differences between orbital graphs and TODs. The close relationship, in this case, between the TOD G and its canonical double covering, is also investigated. The paper contains several examples intended to make these new concepts and results more clear.