5-cycle systems

Abstract

A k-cycle system of a multigraph G is an ordered pair (V,C) where V is the vertex set of G and C is a set of k-cycles, the edges of which partition the edges of G. A k-cycle system of lambda K_v is known as a lambda-fold k-cycle system of order v. A k-cycle system (V,C) of lambda K_v is said to be enclosed (embedded) in a k-cycle system (V\cup U,P) of (lambda+m) K_{v+u} if C \subset P and u,m >= 1 (m=0 and u >= 1). We settle the enclosing problem for lambda-fold 5-cycle systems when u=1 or 2. We settle the embedding problem for lambda-fold 5-cycle systems except possibly in two cases. Other analogues of this are considered and consequently settled.