Existence of almost resolvable directed 5-cycle systems

A directed k-cycle system of order n is a pair ( S, T ), where S is an n-set and T is a collection of arc disjoint directed k-cycles that partition the complete directed graph Kn *. An almost parallel class with deficiency x is a set of directed k-cycles which form a partition of S \ {x}. An almost resolvable directed k-cyclc system is a directed k-cyclc system in which the cycles can be partitioned into almost parallel classes. It is clear that n == 1 (mod k) is a necessary condition for the existence of such a system. It is well known that for k 3 and 4 the necessary condition is also sufficient. In this paper, we shall introduce a special kind of skew Room frames and discuss their constructions. As an application, we show that an almost resolvable directed 5-cycle system of order n exists if and only ifn 1 (mod 5 ).