Small embeddings for partial cycle systems of odd length

Let V(G) and E(G) denote the vertex and edge sets of a graph G respectively. Let Z, = (0, 1, . . . , n l}. Let K,, be the complete graph on n vertices. An m-cycle is a simple graph with m vertices, say uo, . . . , u,__~ in which the only edges are uou,_i and the edges joining ui to Ui+l (for 0 s i 6 m 2). We represent this cycle by (uo, . . . , u,_J or (uo, u,_~, u,_~, . . . , ul) or any cyclic shift of these. A (partial) m-cycle system is an ordered pair (V, C(m)) where C(m) is a set of edge-disjoint m-cycles which partition (a subset of) the edge set of the complete graph with vertex set V. A partial m-cycle system (Z,, C,(m)) is embedded in an m-cycle system (Z,, C,(m)) if C,(m) G C,(m). A natural problem then is to find as small a value of v as possible so that every partial m-cycle system on n vertices can be embedded in an m-cycle system on v vertices. The best result to date is Wilson’s theorem [ll] which shows that all partial m-cycle systems can be finitely embedded, but the size v of the m-cycle system is an exponential function of n. (Of course, Wilson’s result is proved for the embedding of partial graph decompositions in general, not just for m-cycle systems.) The only other results related to this problem deal with the particular case when m = 3. A 3-cycle system is more commonly known as a Steiner triple system. Originally, a finite embedding of a partial Steiner triple system on n vertices in a Steiner triple system on u vertices was found by Treash [lo], but v is an exponential function of