## Abstract

The Ramsey number r(C_k, C_k, C_k), denoted as r_3(C_k), is the smallest positive integer n such that any edge coloring with three colors of the complete graph on n vertices must contain at least one monochromatic cycle C_k. In this project, most literature on the Ramsey numbers r_3(C_k) are overviewed. Algorithms to check if a graph G contains any specific path or cycle and to construct extremal graphs for cycle C_k are developed. All good 3-colorings of complete graph K_10 are constructed to verify the value of Ramsey number r_3(C_4). Ramsey number value of r_3(C_3) is verified by direct point by point extension algorithm. The lower bounds for the Ramsey numbers r_3(C_5), r_3(C_6), and r_3(C_7) are provided as well. Additionally, the possibility of further research for larger k, especially for r_3(C_8) and r_3(C_10) is searched. Most of the results are based on computer algorithms.

## Publication Date

2006

## Document Type

Master's Project

## Student Type

Graduate

## Department, Program, or Center

Computer Science (GCCIS)

## Advisor

Radziszowski, Stanislaw - Chair

## Advisor/Committee Member

Anderson, Peter

## Advisor/Committee Member

Teredesai, Ankur

## Recommended Citation

Li, Yan, "The Study of Ramsey numbers r(C_k, C_k, C_k)" (2006). Thesis. Rochester Institute of Technology. Accessed from

https://repository.rit.edu/theses/6895

## Campus

RIT – Main Campus

## Comments

Note: imported from RIT’s Digital Media Library running on DSpace to RIT Scholar Works in February 2013.