Perfect octagon quadrangle systems

An octagon quadrangle is the graph consisting of an 8-cycle ( x 1 , … , x 8 ) with two additional chords: the edges { x 1 , x 4 } and { x 5 , x 8 } . An octagon quadrangle system [ O Q S ] 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 λ K v defined on X . An octagon quadrangle system Σ = ( X , B ) of order v and index λ is strongly perfect if the collection of all the inside 4-cycle and the collection of all the outside 8-cycle quadrangles, contained in the octagon quadrangles, form a μ -fold 4-cycle system of order v and a ϱ -fold 8-cycle system of order v , respectively. More generally, C 4 -perfect O Q S s and C 8 -perfect O Q S s are also defined. In this paper, following the ideas of polygon systems introduced by Lucia Gionfriddo in her papers [4–7], we determine completely the spectrum of strongly perfect O Q S s , C 4 -perfect O Q S s and C 8 -perfect O Q S s , having the minimum possible value for their indices.