A small embedding for large directed even cycle systems

A directed m-cycle system is a collection of directed edges of the form {(XbX2), (X2' X3)(X3, X4), ... , (Xm-l, xm), (xm' xd}, where Xl, X2, X3, ... , Xm are distinct. A partial directed m-cycle system of order n is a pair (8, C), where C is a collection of edge disjoint m-cycles of the complete directed graph Dn with vertex set S. If the cycles in C partition the edge set of Dn we have the definition of a directed m-cycle system. The object of this paper is the proof that for fixed m = 2k and large n, a partial m-cycle of order n can be embedded in an m-cycle system of order approximately (mn)/2.