Structure in Designs, Coverings and Decompositions

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This application is for the renewal of my discovery grant which will facilitate ongoing research activities. It will serve to support HQP, promote collaborative projects and enhance the dissemination of related results. Combinatorial designs and related graph decompositions and factorizations, as well as packings and coverings, provide an ideal way to understand the interaction properties of complex discrete structures, such as networks, a way to investigate the 'interconnectedness' properties of such structures. The central focus of the proposed research program is the investigation of the structure of combinatorial designs and related objects and to consider designs with a particular structure, or lack thereof. Development in this area will provide a deeper understanding of the structure of the objects involved as well as insight into other combinatorial questions. Designs have well-known applications to statistics, coding theory and scheduling. In addition, there are potential applications to such diverse questions as algorithmic efficiency, software and network testing. I have long been involved with finding various kinds of factorizations, initially uniformly and class-uniformly resolvable designs, broadening this work to include more general factorizations, most notably cycle factorizations. I have also considered cases where factorization is not possible. More recently, I have become interested in the related area of Covering Arrays. This project will support my work in all of these areas. The proposal will use state of the art combinatorial techniques as well advanced algorithmic methods to investigate these structures. It will also provide ample opportunity for HQP training in these areas an beyond. The longstanding Oberwolfach problem, introduced by Ringel in the 1960s has received much attention over the years, with several recent advances. The related Hamilton Waterloo problem, where I have had significant success, requires a factorization of the complete graph into a variety of cycle types. One of the objectives of this project is to build on my recent results and continue investigating these and other related problems. Another goal is to undertake a consideration of combinatorial objects which fail to have a particular structure, such as resolvability, or have excess structure, such as Doubly Resolvable Designs Covering arrays have received much interest of late due to their applications in testing, particularly software and network testing. One of the goals of this proposal is to further investigate these objects and further generalisations of them. This includes cases with restricted interaction sets and generalisations to sequence arrays or transitive arrays.