#### Date of Award

2013

#### Degree Type

Dissertation

#### Degree Name

Doctor of Philosophy in Mathematics

#### Department

Mathematics

#### First Advisor

Nancy Eaton

#### Abstract

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 efficient 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*assigned to v have at least a

_{r}*P*in common. This study considers representations in which the host tree is the com- plete graph on n vertices, [

_{q}*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".

#### Recommended Citation

Heissan, Arlene Mia, "Some Results on Graph Representations and Graph Colorings" (2013). *Open Access Dissertations.* Paper 45.

https://digitalcommons.uri.edu/oa_diss/45