Let K be the complete oriented graph on the finite set of vertices A, A family g = {G(a) : a is an element of A) of spanning subgraphs of K is an orthogonal cover provided every arrow of K occurs in exactly one G(a) and for every two elements a, b is an element of A, the graphs G(a) and G(b)(OP) have exactly one arrow in common. Gronau, Gruttmuller, Hartmann, Leck and Leck [H.-D.O.F. Gronau, M. Gruttmuller, S. Hartmann, U. Leck, V. Leck, On orthogonal double covers of graphs, Designs, Codes and Cryptography 27 (2002) 49-91] have observed that if A has the structure of a finite ring and if f is an element of A is such that both f + 1 and f - 1 are units, then the family, obtained by taking for Go the multiplication graph off and for G(a), the rotation of G(0) by a, defines an orthogonal cover on K. In this article we assume that A 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 A is cyclic or elementary abelian.
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