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.
Department, Program, or Center
Computer Science (GCCIS)
Radziszowski, Stanislaw - Chair
Li, Yan, "The Study of Ramsey numbers r(C_k, C_k, C_k)" (2006). Thesis. Rochester Institute of Technology. Accessed from
RIT – Main Campus