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

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