Shardul Rao

Abstract

Ramsey Theory studies conditions when a combinatorial object contains necessarily some smaller given objects. The role of Ramsey Numbers is to quantify some of the general existential theorems in Ramsey Theory. The objective of this thesis is to try to improve the lower bounds of Ramsey Numbers, in particular the bounds for multi-color graph Ramsey Numbers. Let Gi,G2,---,Gm be graphs on some n. R(G\,G2, ,Gm) denotes the m-color Ramsey number for graphs, avoiding G, in color i for 1 < i < m. Thus, to show R(G\, G2, . , Gm) > N, we need to find an edge coloring for a Kn graph using m colors (for N as large as possible) avoiding G, in color i. An order-based Genetic Algorithm (GA) combined with a greedy coloring heuristic is used as a search heuristic in finding the coloring. Each chromo some is a permutation representing the order in which the edges of graph are colored. (This approach was far more successful than a string representing the coloring.) The algorithm was successful. The known lower bounds for R(C4,C4,C4), R(C4,C4, K3), R(C4,K3, K3), R(C5,C5, C5) were matched, and two new lower bounds for Ramsey numbers, R(C4,C4,K3,K3) > 25 and R(Cs, C4, K3) > 13, were found. *See thesis for correct equations and mathematical terms

Library of Congress Subject Headings

Ramsey numbers; Genetic algorithms; Combinatorial optimization

Publication Date

1997

Document Type

Thesis

Department, Program, or Center

Computer Science (GCCIS)

Advisor

Radziszowski, Stanislaw

Advisor/Committee Member

Anderson, Peter

Advisor/Committee Member

Wolf, Walter

Comments

Note: imported from RIT’s Digital Media Library running on DSpace to RIT Scholar Works. Physical copy available through RIT's The Wallace Library at: QA166 .R36 1997

Campus

RIT – Main Campus

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