Steiner triple systems and cycle structure

Abstract

Since 1847 when Rev. T.P. Kirkman published his first paper [24], research in Steiner triple systems has grown steadily. With increased and ready access to more powerfiul computers in recent times, this growth has accelerated significantly. Problems are now being solved which were previously considered too 'big' to contemplate using pure Mathematical techniques. Cycle structure is an area of Steiner triple systems which has been largely unexplored and was chosen for this thesis as it offers an opportunity to develop and expand knowledge by applying a mixture of computational and theoretical techniques.
For each pair of elements of an STS(v) it is possible to define a cycle list comprising a set of cycles. The length n, of an element of a cycle list is necessarily greater than or equal to 4 and less than or equal to (v-3). The cycle structure of an STS(v) is the collection of all vC 2 cycle lists for that STS(v).
An operation that can be carried out on a Steiner triple system is that of cycle switching. Switching an n-cycle in one Steiner triple system will produce a new Steiner triple system that is not usually isomorphic to the original system. The operation of cycle switching is investigated and the survey of 4-cycle switching given by P. Gibbons[12] for the 80 STS(15)s is extended to 6, 8 and 12-cycles.
Various well known recursive constructions such as the 'doubling construction' and the 'product construction' produce Steiner triple systems based upon previously given Steiner triple systems. An analysis of these constructions and their effect on the cycle structure of the Steiner triple system produced is considered in this thesis. From this work it is now possible, by knowing the cycle structure of the given system, to predict the cycle structure of the produced Steiner triple system. This is of significant use in producing Steiner triple systems lacking certain forbidden configurations.
Perfect and uniform Steiner triple systems may be defined as systems having highly regular cycle structures of two closely related types. Prior to this thesis the total knowledge of perfect and uniform Steiner triple systems was scant and in the case of perfect systems limited to only four values of v. This thesis presents a new technique that produces both perfect and uniforni Steiner triple systems. This technique increases the total number of known perfect Steiner triple systems from four to thirteen and is only limited here by the large size of the systems being produced.
A fi.irther area of considerable development in this thesis is the production of previously unknown equivalence classes of Steiner triple systems. This has been achieved by using a combination of cycle switching, an appropriate construction methodology and computer analysis techniques.
This thesis is a comprehensive study of the cycle structures of Steiner triple systems and documents some significant new results and techniques that will be of use to anyone interested in Steiner triple systems.