Combinatorial Designs, Graphs, and Networks

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Dr. Pike's research group tackles problems in the domains of combinatorial design theory and graph theory. Graphs are mathematical abstractions that represent the core elements of networks, namely their nodes and links between them. Graph theoretic models of networks provide a unified context in which to study diverse network problems. Problems of interest include inhibiting the spread of misinformation through social networks, and how to isolate malware in a computer network. Combinatorial designs are mathematical structures that show how to form a collection of subsets from a master set so that elements occur together a prescribed number of times among the subsets. Designs have natural applications in scheduling. For example, a type of design called a Steiner triple system can be used by a teacher who wants to assign each student to several team projects over the course of the school year so that each team has 3 students and also so that each pair of students works together in a team exactly once. Designs also have applications in coding theory and cryptography. We will study designs with strong structural properties, and which in turn correspond to scheduling scenarios with extra constraints imposed on them. We also intend to prove or refute a conjecture by Cioaba et al. that every connected strongly regular graph of even order has a Class 1 edge colouring; the remaining open case of this conjecture is for block intersection graphs of designs. One of the main goals of this proposal is to provide advanced training opportunities for students, who will acquire experience in mathematical problem solving, technical writing, and programming with parallel computing. Students will be prepared to compete in the job market and contribute to the Canadian economy.