On Cayley line digraphs Original Research Article

Given a colouring Δ of a d-regular digraph G and a colouring Π of the symmetric complete digraph on d vertices with loops, the uniformly induced colouring LΠΔ of the line digraph LG is defined. It is shown that the group of colour-preserving automorphisms of (LG, LΠΔ) is a subgroup of the group of colour-permuting automorphisms of (G, Δ). This result is then applied to prove that if (G, Δ) is a d-regular coloured digraph and (LG, LΠΔ) is a Cayley digraph, then (G, Δ) is itself a Cayley digraph Cay (Ω, Δ) and Π is a group of automorphisms of Ω. In particular, a characterization of those Kautz digraphs which are Cayley digraphs is given. If d=2, for every arc-transitive digraph G, LG is a Cayley digraph when the number k of orbits by the action of the so-called Rankin group is at most 5. If k ⩾ 3 the arc-transitive k-generalized cycles for which LG is a Cayley digraph are characterized.