Perfect dodecagon quadrangle systems

A dodecagon quadrangle   is the graph consisting of two cycles: a 12-cycle (x1,x2,…,x12)(x1,x2,…,x12) and a 4-cycle (x1,x4,x7,x10)(x1,x4,x7,x10). A dodecagon quadrangle system   of order nn and index ρρ [ DQS] is a pair (X,H)(X,H), where XX is a finite set of nn vertices and HH is a collection of edge disjoint dodecagon quadrangles (called blocks  ) which partitions the edge set of ρKnρKn, with vertex set XX. A dodecagon quadrangle system   of order nn is said to be perfect   [PDQS] if the collection of 4-cycles contained in the dodecagon quadrangles form a 4-cycle system of order nn and index μμ. In this paper we determine completely the spectrum of DQSs of index one and of PDQSs with the inside 4-cycle system of index one.