| MSC: | 5B15 | |
| 5B40,5E20 |
Abstract:
Let K be the complete oriented graph on the finite set of
vertices A. A family G = {Ga : a ∈ A} of spanning subgraphs of K
is an orthogonal cover provided every arrow of K occurs in exactly
one Ga and for every two elements a, b ∈ A, the graphs Ga and
Gop
b have exactly one arrow in common. Gronau, Grüttmüller,
Hartmann, Leck and Leck (2002) have observed that if A has the
structure of a finite ring and if f ∈ A is such that both f + 1
and f − 1 are units, then the family, obtained by taking for G0
the multiplication graph of f and for Ga the rotation of G0 by a,
defines an orthogonal cover on K. In this manuscript 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.