ON CAYLEY LINE DIGRAPHS

Given a colouring Delta of a d-regular digraph G and a colouring Pi of the symmetric complete digraph on d vertices with loops, the uniformly induced colouring L(Pi)Delta of the line digraph LG is defined. It is shown that the group of colour-preserving automorphisms of (LG,L(Pi)Delta) is a subgroup of the group of colour-permuting automorphisms of (G, Delta). This result is then applied to prove that if (G,Delta) is a d-regular coloured digraph and (LG,L(Pi)Delta) is a Cayley digraph, then (G,Delta) is itself a Cayley digraph Cay (Omega,Delta) and Pi is a group of automorphisms of Omega. In particular, a characterization of those Kautz digraphs which are Cayley digraphs is given. If d=2, for every are-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 greater than or equal to 3 the are-transitive k-generalized cycles for which LG is a Cayley digraph are characterized.