## Abstract

The two-color Ramsey number R(G , H) is defined to be the smallest integer n such that any graph F on n vertices contains either a subgraph isomorphic to G or the complement of F contains a subgraph isomorphic to H. Ramsey numbers serve to quantify many of the existing theorems of Ramsey theory, which looks at large combinatorial objects for certain given smaller combinatorial objects that must be present. In 1989 George R. T. Hendry presented a table of two-color Ramsey numbers R(G , H) for all pairs of graphs G and H having at most five vertices. This table left seven unsolved cases, of which three have since been solved. This thesis eliminates one of the remaining four cases, R(W5 , K5), where a K5 is the complete graph on five vertices and a W5 is a wheel of order 5, which can be pictured as a wheel having four spokes or as a cycle of length 4 having all four vertices adjacent to a central vertex. In this thesis we show R(W5, K5) to be equal to 27, utilizing a combinatorial approach with significant computations. Specifically we use a technique developed by McKay and Radziszowski to effectively glue together smaller graphs in an effort to prove exhaustively that no graph having 27 vertices exists that does not contain an independent set on five vertices or a subgraph isomorphic to W5. The previous best bounds for this case were 27 <= R( W_5 , K_5 ) <= 29.

## Library of Congress Subject Headings

QA166 .S75 2004; Ramsey numbers; Ramsey theory; Graph theory; Combinatorial analysis

## 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

Hemaspaandra, Edith

## Advisor/Committee Member

Tse, Kung-Kuen

## Recommended Citation

Stinehour, Joshua, "R(W5 , K5) = 27" (2006). Thesis. Rochester Institute of Technology. Accessed from

https://repository.rit.edu/theses/6897

## Campus

RIT – Main Campus

## Comments

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