Hexagon quadrangle systems

A hexagon quadrangle system   of order nn and index ρρ [HQSρ(n)HQSρ(n)] is a pair (X,H)(X,H), where XX is a finite set of n   vertices and HH is a collection of edge disjoint hexagon quadrangles (called blocks  ) which partitions the edge set of ρKnρKn, with vertex set XX. A hexagon quadrangle system is said to be a 4-nesting  [N(4)-HQS][N(4)-HQS] if the collection of all the 4-cycles contained in the hexagon quadrangles is a ρ/2ρ/2-fold 4-cycle system. It is said to be a 6-nesting  [N(6)-HQS][N(6)-HQS] if the collection of 6-cycles contained in the hexagon quadrangles is a (3ϱ4)-fold 6-cycle system. It is said to be a (4,6)(4,6)-nesting  , briefly a N(4,6)-HQSN(4,6)-HQS, if it is both a 4-nesting and a 6-nesting.In this paper we determine completely the spectrum of N(4,6)-HQSN(4,6)-HQS for λ=6hλ=6h, μ=4hμ=4h and ρ=8hρ=8h, h positive integer.