Date of Award

2019

Document Type

Thesis

Degree Name

Bachelors

Department

Natural Sciences

First Advisor

Poimenidou, Eirini

Area of Concentration

Mathematics

Abstract

We consider graph decompositions and odd coverings of complete graphs. Chapter 1 is dedicated to an introduction to graph theory including basic, although essential, definitions. In Chapter 2, we consider the Oberwolfach problem, which asks if the complete graph K2n+1 can be decomposed into isomorphic copies of any 2-factor of K2n+1. We develop a method of constructing solutions to infinitely many cases of the Oberwolfach problem when a 1-rotational solution is given. In Chapter 3, motivated by decomposition and the Oberwolfach problem, we develop the concept of an odd covering, which to our knowledge has not been considered much, if it all, in the literature. We show that there is an odd covering of K2n+1 into isomorphic 2-factors, and we determine when there exists an odd covering of K2n+1 into copies of a graph with an end vertex. We also show when there exists an odd covering of K2n+1 into copies of a given cycle. We conclude the thesis with some conjectures and open problems related to odd coverings.

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