On Cayley line digraphs

Abstract

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 L Π Δ of the line digraph LG is defined. It is shown that the group of colour-preserving automorphisms of ( LG, L Π Δ) 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, L Π Δ) 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.