The Doyen-Wilson Theorem Extended to 5-Cycles

Let ~m = {0, 1 . . . . , m 1}. A n m-cycle is a graph (v0, V l , . . . , u~_ 1) with vertex set {viii ~ E m} and edge set {{vi, Vi+a}]i ~ 7/m} (reducing the subscript modulo m). A n m-cycle system of a graph G is an o rdered pair (V, C), where V is the vertex set of G and C is a set of m-cycles, the edges of which part i t ion the edges of G. A n m-cycle system (of order n) is an m-cycle system of K n. A partial m-cycle system of order n is an m-cycle system of a subgraph of Kn. In this paper, we are interested in the embedding problem for m-cycle systems. A (partial) m-cycle system (V, C) is said to be embedded in the m-cycle system (W, P ) if V c W and C c P. In 1973, Doyen and Wilson obta ined the following result for embedding 3-cycle systems, also known as Steiner triple systems.