## Mathematics Dissertations

8-13-2019

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics and Statistics

Zhongshan Li

Guantao Chen

#### Abstract

Given a (multi)graph, the density is defined by $\Gamma(G)=\max \Big\{\frac{2|E(U)|}{|U|-1}:\,\, U \subseteq V, \,\, |U|\ge 3 \hskip 2mm {\rm and \hskip 2mm odd} \Big\}.$ The {\bf chromatic index} $\chi'(G)$ of a graph $G$ is the minimum number of colors that required to color the edges of $G$ such that two adjacent edges receive different colors. It is known that $\chi'(G)\geq \Gamma(G)$. The {\bf cover index} $\xi(G)$ of $G$ is the greatest integer $k$ for which there is a coloring of $E$ with $k$ colors such that each vertex of $G$ is incident with at least one edge of each color. A sign pattern is a matrix whose entries are from the set $\{+, -, 0\}$.

In part 1, we will generally discuss the connections between the density and the chromatic index. In particular, the Goldberg-Seymour conjecture states that $\chi'(G)=\lceil\Gamma(G)\rceil$ if $\chi'(G)>\Delta+1$, where $\Delta$ is the maximum degree of $G$. Some open problems are mentioned at the end of part 1. In particular, a dual conjecture to the Goldberg-Seymour conjecture on the cover index is discussed. A proof of the Goldberg-Seymour conjecture is given In part 2.

In part 3, we will present a connection between the minimum ranks of sign pattern matrices and point-line configurations.

#### DOI

https://doi.org/10.57709/14772563