Consistent Cycles in Graphs and Digraphs

Let Γ be a finite digraph and let G be a subgroup of the automorphism group of Γ. A directed cycle ~ C of Γ is called G-consistent whenever there is an element of G whose restriction to ~ C is the 1-step rotation of ~ C. Consistent cycles in finite arc-transitive graphs were introduced by Conway in one of his public lectures. He observed that the number of G-orbits of G-consistent cycles of an arc-transitive group G is precisely one less than the valency of the graph. In this paper, we give a detailed proof of this result in a more general settings of arbitrary groups of automorphisms of graphs and digraphs.