Combinatorial problems on Abelian Cayley graphs

This dissertation describes and solves various combinatorial problems on Abelian Cayley Graphs. Specifically, we address two different problems. First, we address finding Hamiltonian decompositions in Abelian Cayley Graphs, using a technique that takes an Abelian Cayley Graph with a Hamiltonian decomposition and then constructs a Hamiltonian decomposition in a larger Abelian Cayley graph which has many embedded copies of the smaller graph. Second, we address classifying circulant covers over circulant graphs. A circulant graph is a Cayley graph over a finite cyclic (hence, Abelian) group. The motivating question here is to classify when a circulant cover of a circulant base graph must have a covering map which is isomorphic to a group homomorphism, without relabelling the group elements. We give a complete classification of covers in the case where the circulants are valency 3, and present general techniques to study covers between circulants of higher valency.