| 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. |
