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    Octagon Quadrangle Systems nesting 4-kite-designs having equi-indices

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    An octagon quadrangle is the graph consisting of an 8-cycle (x1,x2,,x8)(x_1,x_2,\dots,x_8)(x1,x2,…,x8) with two additional chords: the edges {x1,x4}\{x_1,x_4\}{x1,x4} and {x5,x8}\{x_5,x_8\}{x5,x8}. An octagon quadrangle system of order vvv and index λ\lambdaλ (OQS) is a pair (X,H)(X,H)(X,H), where XXX is a finite set of vvv vertices and HHH is a collection of edge disjoint octagon quadrangles (called blocks) which partition the edge set of λKv\lambda K_vλKv defined on XXX. An octagon quadrangle system Σ=(X,H)\Sigma=(X,H)Σ=(X,H) of order vvv and index λ\lambdaλ is said to be upper C4C_4C4-perfect if the collection of all of the upper 4-cycles contained in the octagon quadrangles form a μ\muμ-fold 4-cycle system of order vvv; 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\muμ-fold 4-cycle system of order vvv and also the collection of all of the outside 8-cycles contained in the octagon quadrangles form a ρ\rhoρ-fold 8-cycle system of order vvv. In this paper, the authors determine the spectrum for these systems, in the case that it is the largest possible
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