Date of Award

Summer 2011

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Computational Analysis and Modeling

First Advisor

Galen Turner

Abstract

The Ramsey number R(ω, α) is the minimum number n such that every graph G with |V(G)| ≥ n has an induced subgraph that is isomorphic to a complete graph on ω vertices, Kω, or has an independent set of size α, Nα. Graphs having fewer than n vertices that have no induced subgraph isomorphic to K ω or Nα form a class of Ramsey graphs, denoted ℜ(ω, α). This dissertation establishes common structure among several classes of Ramsey graphs and establishes the complete list of ℜ(3, 4).

The process used to find the complete list for ℜ(3, 4) can be extended to find other Ramsey numbers and Ramsey graphs. The technique for finding a complete list for ℜ(ω, α), a) is inductive on n vertices in that a complete list of all graphs in ℜ(ω, α) having exactly n vertices can be used to find the complete list n + 1 vertices. This process can be repeated until any extension is not in ℜ(ω, α), and thus R(ω, α) has been determined. We conclude by showing how to extend methods presented in proving R(3, 4) in finding R(5, 5).

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