#### Date of Award

6-2015

#### Degree Name

Doctor of Philosophy

#### Department

Mathematics

#### First Advisor

Dr. Ping Zhang

#### Second Advisor

Dr. Gary Chartrand

#### Third Advisor

Dr. Allen Schwenk

#### Fourth Advisor

Dr. Heather Jordon

#### Keywords

Edge coloring, Twin edge coloring, K-Ramsey, Domination, Matching, Graphs, Mathematics applications

#### Abstract

Edge colorings have appeared in a variety of contexts in graph theory. In this work, we study problems occurring in three separate settings of edge colorings.

For more than a quarter century, edge colorings have been studied that induce vertex colorings in some manner. One research topic we investigate concerns edge colorings belonging to this class of problems. By a twin edge coloring of a graph *G* is meant a proper edge coloring of *G* whose colors come from the integers modulo k that induce a proper vertex coloring in which the color of a vertex is the sum of the colors of its incident edges. The minimum *k* for which *G* has a twin edge coloring is the twin chromatic index of *G*. Several results on this concept have been obtained as well as a conjecture.

A red-blue coloring of a graph *G* is an edge coloring of *G* in which every edge is colored red or blue. The Ramsey number of *F* and *H* is the smallest positive integer n such that every red-blue coloring of the complete graph of order *n* results in a red *F* or a blue *H*. The related concept of bipartite Ramsey number has been defined and studied when *F* and *H* are bipartite. We introduce a new class of Ramsey numbers which extend these two well-studied concepts in the area of extremal graph theory and present results and problems on these new concepts.

Let *F* be a graph of size 2 or more having a red-blue coloring in which there is at least one edge of each color. One blue edge is designated as the root of *F*. For such an edge- colored graph *F*, an *F*-coloring of a graph *G* is a red-blue coloring of *G* in which every blue edge is the root of some copy of *F* in *G*. The F-chromatic index of *G* is the minimum number of red edges in an *F*-coloring of *G*. In this setting, we provide a bichromatic view of two well-known concepts in graph theory, namely matchings and domination, and present results and problems in this area of research.

#### Access Setting

Dissertation-Open Access

#### Recommended Citation

Johnston, Daniel, "Edge Colorings of Graphs and Their Applications" (2015). *Dissertations*. 586.

https://scholarworks.wmich.edu/dissertations/586