## Mathematics Dissertations

8-10-2021

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics and Statistics

Guantao Chen

#### Abstract

Graph edge coloring is a well established subject in the field of graph theory. It is one of the basic combinatorial optimization problem: Color the edges of a graph $G$ with as few colors as possible such that each edge receives a color and adjacent edges receive different colors. The minimum number of colors needed for such a coloring of $G$ is called the chromatic index, denoted by $\chi'(G)$. Let $\Delta(G)$ and $\mu(G)$ be maximum degree and maximum multiplicity of $G$, respectively. Vizing and Gupta, independently, proved in the 1960s that $\chi'(G)\le\Delta(G)+\mu(G)$, by using the Vizing fan as main tool. Vizing fans and Vizing's Theorem play an important role in graph edge coloring. In this dissertation, we introduce two new generalizations of Vizing fans and obtain their structural properties for simple graphs, and partly comfirm one conjecture on the precoloring extension of Vizing's Theorem for multigraphs.

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