Structure in designs, coverings and decompositions

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Combinatorial Designs 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; 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 and numerical analysis. I have long been involved with finding various kinds of factorizations, initially uniform 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 Covering Arrays. This project will support my work in all of these areas. The longstanding Oberwolfach problem, introduced by Ringel in the 1960s as a seating problem has recieved much attention over the years, with several recent advances. The more general Hamilton-Waterloo problem requires a factorization of the complete graph into a variety of cycle types. One of the objectives of this project is to continue investigation into these problems. Another goal is to undertake a consideration of combinatorial objects which fail to have a particular structure, such as resolvability. 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, particularly with respect to substructures useful for testing that they may exhibit. This includes, mixed strength covering arrays and covering arrays with forbidden pairs or other configurations as well as the development, together with collaborators, of an online resource for generation of covering arrays, an important industrial resource.