Perfect Octagon Quadrangle Systems with upper C(4)-systems and a large spectrum

An octagon quadrangle is the graph consisting of an 8-cycle (x1,x2,…,x8) with two additional chords: the edges {x1,x4} and {x5,x8}. An octagon quadrangle system of order v and index λ (OQS) is a pair (X,H), where X is a finite set of v vertices and H is a collection of edge disjoint octagon quadrangles (called blocks) which partition the edge set of $\lambda λKv defined on X. An octagon quadrangle system Σ=(X,H) of order v and index λ is said to be upper C4-perfect if the collection of all of the upper 4-cycles contained in the octagon quadrangles form a μ-fold 4-cycle system of order v; it is said to be upper strongly perfect if the collection of all of the upper 4-cycles contained in the octagon quadrangles form a μ-fold 4-cycle system of order v and also the collection of all of the outside 8-cycles contained in the octagon quadrangles form a ρ-fold 8-cycle system of order v. In this paper, the authors determine the spectrum for these systems, in the case that it is the largest possible.