Perfect octagon quadrangle systems with upper C4-systems

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 ρ [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 ρ K v 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 upper4-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 upper4-cycles contained in the octagon quadrangles form a μ -fold 4-cycle system of order v and also the collection of all of the outside8-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.