Date of Award

8-1-2020

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics and Statistics

First Advisor

Guantao Chen

Second Advisor

Zhongshan Li

Third Advisor

Songling Shan

Fourth Advisor

Hendricus Van der Holst

Fifth Advisor

Yi Zhao

Abstract

The graph edge coloring problem is to color the edges of a graph such that adjacent edges receives different colors. Let $G$ be a simple graph with maximum degree $\Delta$. The minimum number of colors needed for such a coloring of $G$ is called the chromatic index of $G$, written $\chi'(G)$. We say $G$ is of class one if $\chi'(G)=\Delta$, otherwise it is of class 2. A majority of edge coloring papers is devoted to the Classification Problem for simple graphs. A graph $G$ is said to be \emph{overfull} if $|E(G)|>\Delta \lfloor |V(G)|/2\rfloor$. Hilton in 1985 conjectured that every graph $G$ of class two with $\Delta(G)>\frac{|V(G)|}{3}$ contains an overfull subgraph $H$ with $\Delta(H)=\Delta(G)$. In this thesis, I will introduce some of my researches toward the Classification Problem of simple graphs, and a new approach to the overfull conjecture together with some new techniques and ideas.

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