Orthogonal covers by multiplication graphs

Let KK be the complete oriented graph on the finite set of vertices AA. A family G={Ga:a∈A}G={Ga:a∈A} of spanning subgraphs of KK is an orthogonal cover   provided every arrow of KK occurs in exactly one GaGa and for every two elements a,b∈Aa,b∈A, the graphs GaGa and Gbop have exactly one arrow in common. Gronau, Grüttmüller, Hartmann, Leck and Leck [H.-D.O.F. Gronau, M. Grüttmüller, S. Hartmann, U. Leck, V. Leck, On orthogonal double covers of graphs, Designs, Codes and Cryptography 27 (2002) 49–91] have observed that if AA has the structure of a finite ring and if f∈Af∈A is such that both f+1f+1 and f−1f−1 are units, then the family, obtained by taking for G0G0 the multiplication graph of ff and for GaGa the rotation of G0G0 by aa, defines an orthogonal cover on KK. In this article we assume that AA is a finite abelian group and proceed to (i)generalize this construction to arbitrary endomorphisms of the underlying group and describe the possible graphs,(ii)introduce a duality on the set of orthogonal covers and(iii)give detailed descriptions of the covers in the case where AA is cyclic or elementary abelian.