Perfect dodecagon quadrangle systems

A dodecagon quadrangle is the graph consisting of two cycles: a 12-cycle (x1, x2, ..., x12) and a 4-cycle (x1, x4, x7, x10). 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 ρKn, 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. 1 In memory of Lucia All the results contained in this paper were obtained by Lucia. She completed this research during the last week of February of 2008. This is Lucia’s last paper. Her intention and wish was to give a talk on this subject at Combinatorics 2008, in June 2008, in Costermano (Italy). However, Lucia’s life changed dramatically: on March 17, after medical treatment in Milan, where she usually went since 2006, her oncologist communicated to her that she had only a few months to live. From that moment, she passed through extremely difficult period and terrible moments. After a period of chemoterapy and radioterapy, which she faced with great courage, after a period of unbelievable sufferings, she passed away on July 21. Lucia was only 34 years old. I, my wife and my son Giuseppe, will never forget the last four months of her life, full of great sadness and sorrow. Lucia’s splendid smile will remain with us forever. (Mario Gionfriddo)