Nonagon quadruple systems: existence, balance, embeddings

A cycle of length 9 of vertices (x1, x2, . . . , x9), in the cyclical order, with the three edges {x1, x4}, {x4, x7}, {x1, x7} is called an NQ-graph or also a nonagon quadruple graph. A nonagon quadruple system, briefly NQS, of order v and index λ is an NQ-decomposition of the complete multigraph λKv. An NQS is said to be perfect if the inside K3, generated by the vertices x1,x4,x7, forms a Steiner triple system; it is said to be balanced if all the vertices have the same degree. In this paper, the spectrum of NQSs, the spectrum of perfect NQSs and the spectrum of balanced NQSs are completely determined.