Steiner pentagon systems

A Steiner pentagon system is a pair (K_n, P) where K_n isthe complete undirected graph on n vertices. P is a collection of edge-disjoint pentagons which partition K_n, and such that every part of distinct vertices of K_n is joined by a path of length two in exactly one pentagon of the collection P. The number n is called the order of the system. This paper gives a somplete solution of the existence problem of Steiner pentagon systems. In particular it is shown that the spectrum for Steiner pentagon systems (=the set of all orders for which a Steiner pentagon system exists) is precisely the set of all n ≡ 1 or 5 (mod 10), except 15, for which no such system exists.