#### Title

#### Date of Award

4-1-2023

#### Degree Name

Doctor of Philosophy

#### Department

Mathematics

#### First Advisor

Ping Zhang, Ph.D.

#### Second Advisor

Gary Chartrand, Ph.D.

#### Third Advisor

John Martino, Ph.D.

#### Fourth Advisor

Ebrahim Salehi, Ph.D.

#### Keywords

Cycle rank, Dutch windmill, four color theorem, graph theory, labelings, zonality

#### Abstract

Graph labeling and coloring are among the most popular areas of graph theory due to both the mathematical beauty of these subjects as well as their fascinating applications. While the topic of labeling vertices and edges of graphs has existed for over a century, it was not until 1966 when Alexander Rosa introduced a labeling, later called a graceful labeling, that brought the area of graph labeling to the forefront in graph theory. The subject of graph colorings, on the other hand, goes back to 1852 when the young British mathematician Francis Guthrie observed that the countries in a map of England could be colored with four colors so that every two adjacent countries are colored differently. This led to the Four Color Problem, which is the problem of determining whether the regions of every plane map can be colored with four or fewer colors in such a way that every two adjacent regions are colored differently. A computer aided solution for the Four Color Problem was announced in 1976 by Kenneth Appel and Wolfgang Haken, resulting in the famous Four Color Theorem.

In 2014, the Australian physicist Cooroo Egan introduced a graph labeling referred to as a zonal labeling. A zonal labeling is a vertex labeling of a connected plane graph G with the two nonzero elements of the ring Z3 of integers modulo 3 such that the sum of the labels of the vertices on the boundary of every region of G is the zero element of Z3. A graph possessing a zonal labeling is a zonal graph. A related labeling, called an inner zonal labeling, is a labeling of the vertices of a plane graph G with the nonzero elements of Z3 such that the sum of the labels of the vertices on the boundary of every interior region of G is the zero element of Z3. A graph possessing an inner zonal labeling is an inner zonal graph. There is a close connection between the existence of zonal and inner zonal labelings of planar graphs and the Four Color Theorem. In this work, we study zonality and inner zonality for several well-known classes of graphs, determine which of these graphs are zonal or inner zonal, and present characterization results on the structures of zonal graphs. Furthermore, we investigate a relationship between zonal graphs and inner zonal graphs and establish a connection between inner zonal graphs and the Four Color Theorem.

#### Access Setting

Dissertation-Open Access

#### Recommended Citation

Bowling, Andrew, "Zonality in Graphs" (2023). *Dissertations*. 3934.

https://scholarworks.wmich.edu/dissertations/3934