Structural properties and generalizations of combinatorial designs

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The origin of combinatorial designs lies in their application to the design of statistical experiments to test interactions between objects. In cases where objects are to be arranged in a circular manner, certain types of experiments can be modelled by cycle decompositions of graphs, also called cycle designs. In addition, combinatorial designs have applications to areas such as software testing, coding theory and communication systems. I propose to study structural aspects of cycle designs including colourings of the points and blocks of these designs and intersection properties of cycle designs. Colourings of the points of designs have applications in experimental design. Block colourings of designs relate to graph factorizations and the well-known Oberwolfach problem. I aim to construct certain types of cycle designs with desirable colouring properties. In addition, I will further the development of the theory of generalized designs, packings and coverings. Certain cases of these objects model various types of colourings of ordinary designs. Moreover, generalized designs, packings and coverings relate to many existing types of combinatorial designs, so their study will prove a powerful tool to combinatorial design theorists. Finally, I will consider a class of graphs called cycle intersection graphs, which are formed based on the the intersection of cycles in a cycle design. Analogues of these graphs have previously been studied for other classes of designs. One question that will be considered is whether the cycles in a design can be ordered so that consecutive cycles intersect in a particular way.