Date of Award

2017

First Advisor

Aaron Williams

Second Advisor

Amanda Landi

Abstract

Graph theory is an important subject within discrete mathematics and computer science. The subject is focused on the study of graphs which are mathematical structures that are able to model a set of objects called vertices and \related" pairs of objects called edges. The degree of a vertex is the number of edges (i.e. relationships) that it belongs to. A tree is a type of graph that is connected and has no cycles, and a forest is a graph containing one or more trees. One of the most well-studied open problems in graph theory is the graceful tree conjecture. Roughly speaking, a graph has a graceful labeling if its vertices and edges can be assigned integers in an extremely balanced manner, and the conjecture states that every tree has one of these graceful labelings. A cordial labeling is a less stringent version of graceful labeling that is only concerned with the labels 0 and 1, and a graph is cordial if it has such a labeling. The main result of this thesis is the following: A forest is not cordial if and only if every vertex has odd-degree and the number of trees in the forest is 2 modulo 4. This result was obtained as part of a larger theoretical and empirical investigation into the graceful tree conjecture initiated by Myrvold, Panjeer, and Williams at the University of Victoria. This thesis outlines that previous body of work and extends it to the new characterization of cordial forests.

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