#### Date of Award

Spring 2013

#### Degree Type

Thesis

#### Degree Name

Departmental Honors

#### Department

Mathematics and Statistics

#### First Advisor

David E. Brown

#### Abstract

Paired comparison is the process of comparing objects two at a time. A *tournament* in Graph Theory is a representation of such paired comparison data. Formally, an *n-tournament* is an oriented complete graph on *n* vertices; that is, it is the representation of a paired comparison, where the winner of the comparison between objects *x* and *y* (*x* and *y* are called *vertices*)is depicted with an arrow or *arc* from the winner to the other.

In this thesis, we shall prove several results on tournaments. In Chapter 2, we will prove that the maximum number of vertices that can beat exactly *m* other vertices in an *n*-tournament is min{2*m* + 1,2*n* - 2*m* - 1}. The remainder of this thesis will deal with tournaments whose arcs have been colored. In Chapter 3 we will define what it means for a *k*-coloring of a tournament to be *k*-primitive. We will prove that the maximum *k* such that some strong *n*-tournament can be *k*-colored to be *k*-primitive lies in the interval [(^{n-1} _{2}),(^{n} _{2})-[^{n}/_{4}]). In Chapter 4, we shall prove special cases of the following 1982 conjecture of Sands, Sauer, and Woodrow from [14]: Let *T* be a 3-arc-colored tournament containing no 3-cycle *C* such that each arc in *C* is a different color. Then *T* contains a vertex *v* such that for any other vertex *x,x* has a monochromatic path to *v*.

#### Recommended Citation

Mousley, Sarah Camille, "Tournament Directed Graphs" (2013). *Undergraduate Honors Capstone Projects*. 170.

https://digitalcommons.usu.edu/honors/170

#### Included in

*Copyright for this work is retained by the student. If you have any questions regarding the inclusion of this work in the Digital Commons, please email us at .*