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Acknowledgments 
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First, I wish to thank my advisor, Goodrich C. White Professor of Mathematics Emeritus Dr. Ronald J. Gould. This dissertation would not have been possible without his guidance, constant support and patience. I would also like to thank my committee members Dr. Dwight Duffus, Dr. Hao Huang and Dr. Vojt\u{e}ch R\"odl for their careful consideration and helpful suggestions.

I offer thanks to my family and friends for their unending support over the years. To my partner, Patrick Banom, for his love and encouragement. To my father, Damian Fuller, my mother, Lisa Fuller, and my sister, Laura Wray, for their unfailing faith that I would succeed. To my best friend, Katherine St. Clair Misar, for her enthusiastic support and my friend, Laura Gruca, for her understanding. To my nephew, William Wray. 

Finally, I would like to thank all of my Professors and teachers who have helped to guide my path throughout the years and who have encouraged my love of mathematics. In particular, to my undergraduate advisor, Dr. John T. Saccoman, who helped me discover the joy of trying to answer the unanswered questions in Graph Theory. 