A. Granville, H.-D.O.F. Gronau, R. C. Mullin
On a problem of Hering concerning orthogonal covers of $\vec{K}_n$
Preprint series: Preprints aus dem Fachbereich Mathematik, Universität Rostock
MSC:
05A99 None of the above but in this section
Abstract: A Hering configuration of type $k$ and order $n$ is a factorization of
the complete digraph $\vec{K}_n$ into $n$ factors each of wich
consists
of an isolated vertex and the edge-disjoint union of directed
$k$- cycles wich has the additional property that for any pair of
distinct factors say $\vec{G}_i$ and $\vec{G}_j$ there is precisely
one pair of vertices, say $\{a, b\}$ such that $\vec{G}_i$ contains
the
directed edge $(a, b)$ and $\vec{G}_j$ contains the directed edge
$(b, a)$. Clearly a necessary condition for a Hering configuration is
$n\equiv (\mbox{mod } k)$. It is shown here that for any fixed $k$,
this
condition is asymptotically sufficient, and, it is shown to be always
sufficient for $k=4$
Keywords: desings, covers of graph