Enclosings of λ-fold 5-cycle systems for u=2

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 λKv is known as a λ-fold k-cycle system of order v. A k-cycle system of λKv(V,C) is said to be enclosed in a k-cycle system of (λ+m)Kv+u(V∪U,P) if C⊂P and u,m≥1. In this paper the enclosing problem for 5-cycle systems is settled in the general situation where the three parameters λ, m, and v are allowed to vary freely and u is constrained to the difficult case of adding two vertices. New graph theoretic approaches are introduced to handle this situation developing an avenue of research that is of interest in its own right.