Perfect octagon quadrangle systems

An octagon quadrangle is the graph consisting of an 8-cycle (x1,…,x8) with two additional chords: the edges {x1,x4},{x1,x4} and {x5,x8}. An octagon quadrangle system (OQS) of order v and index λ is a pair (X,B), where X is a finite set of v vertices and B is a collection of edge disjoint octagon quadrangles, which partitions the edge set of λK(v) defined on X. An OQS Σ=(X,B) of order v and index λ is strongly perfect if the collection of all the inside 4-cycles contained in the octagon quadrangles form a μ-fold 4-cycle system of order v, and the collection of all the outside 8-cycle quadrangles contained in the octagon quadrangles form a ϱ-fold 8-cycle system of order v. More generally, C4-perfect OQSs and C8-perfect OQSs are also defined. In this paper, following the ideas of polygon systems introduced by L. Gionfriddo in her papers [Bull. Inst. Combin. Appl. 48 (2006), 73-81; MR2259705; Discrete Math. 308 (2008), no. 2-3, 231-241; MR2378021 (2008k:05167); Australas. J. Combin. 36 (2006), 167-176; MR2262617 (2007e:05025); Discrete Math. 309 (2009), no. 2, 505-512; MR2478727 (2010f:05122)], we determine completely the spectrum of strongly perfect OQSs, C4-perfect OQSs and C8-perfect OQSs, having the minimum possible value for their indices.