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 ** of Γ is called G-consistent whenever there is an element of G whose restriction to ** is the 1-step rotation of **. Consistent cycles in finite arc-transitive graphs were introduced by J. H. Conway in his public lectures at the Second British Combinatorial Conference in 1971. 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 setting of arbitrary groups of automorphisms of graphs and digraphs.