Perfect dodecagon quadrangle systems

A dodecagon quadrangle is the graph consisting of two cycles: a 12-cycle ( x 1 , x 2 , … , x 12 ) and a 4-cycle ( x 1 , x 4 , x 7 , x 10 ) . A dodecagon quadrangle system of order n and index ρ [ DQS] is a pair ( X , H ) , where X is a finite set of n vertices and H is a collection of edge disjoint dodecagon quadrangles (called blocks) which partitions the edge set of ρ K n , with vertex set X . A dodecagon quadrangle system of order n is said to be perfect [PDQS] if the collection of 4-cycles contained in the dodecagon quadrangles form a 4-cycle system of order n and index μ . 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.