#### Date Presented

Spring 4-15-2010

#### Document Type

Thesis

#### Degree Name

Bachelor of Science

#### Department

Mathematics

#### First Advisor

Kevin Peterson

#### Second Advisor

Danny Cline

#### Third Advisor

Nancy Cowden

#### Abstract

We study graph colorings of the form made popular by the four-color theorem. Proved by Appel and Haken in 1976, the Four-Color Theorem states that all planar graphs can be vertex-colored with at most four colors. We consider an alternate way to prove the Four-Color Theorem, introduced by Hadwiger in 1943 and commonly know as Hadwiger’s Conjecture. In addition, we examine the chromatic number of graphs which are not planar. More specifically, we explore adding edges to a planar graph to create a non-planar graph which has the same chromatic number as the planar graph which we started from.

#### Recommended Citation

Cates, Sarah E., "The Four-Color Theorem and Chromatic Numbers of Graphs" (2010). *Undergraduate Theses and Capstone Projects*. 77.

https://digitalshowcase.lynchburg.edu/utcp/77

#### Included in

Number Theory Commons, Other Applied Mathematics Commons, Other Mathematics Commons, Other Physical Sciences and Mathematics Commons, Set Theory Commons, Special Functions Commons