### Some Results on Graph Representations and Graph Colorings [thesis]

Arlene Heissan
A graph G(V, E) is a structure used to model pairwise relations between a set of objects. In this context, a graph is a collection of vertices (representing the objects) and a collection of edges (representing the relation) that connect pairs of vertices. It is possible to represent a graph using an adjacency matrix, but often this is not the most efficient representation of the relation. In studying graph representation, the object is to capture the structure of the graph more efficiently
more » ... re efficiently using a variety of other discrete structures. This work considers path representations of graphs. Consider a host graph, H. A path representation [H : r : q] of a target graph G is a labeling in which each vertex is assigned a unique path of length r found in H in such a way that if uv ∈ E(G), then the P r assigned to u and the P r assigned to v have at least a P q in common. This study considers representations in which the host tree is the complete graph on n vertices, [K n , r, q] which will be referred to as P r,q -representations. This work also considers the area in graph theory known as vertex-coloring, specifically coloring planar graphs, and explores a special class of planar graphs called "coils".