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Applications of Graph Theory and Topology to Combinatorial Designs

This dissertation is concerned with the existence and the isomorphism of designs. The first part studies the existence of designs. Chapter I shows how to obtain a design from a difference family. Chapters II to IV study the existence of an affine 3-(p^m,4,λ) design where the v-set is the Galois field GF(p^m). Associated to each prime p, this paper constructs a graph. If the graph has a 1-factor, then a difference family and hence an affine design exists. The question arises of how to determine when the graph has a 1-factor. It is not hard to see that the graph is connected and of even order. Tutte's theorem shows that if the graph is 2-connected and regular of degree three, then the graph has a 1-factor. By using the concept of quadratic reciprocity, this paper shows that if p Ξ 53 or 77 (mod 120), the graph is almost regular of degree three, i.e., every vertex has degree three, except two vertices each have degree tow. Adding an extra edge joining the two vertices with degree tow gives a regular graph of degree three. Also, Tutte proved that if A is an edge of the graph satisfying the above conditions, then it must have a 1-factor which contains A. The second part of the dissertation is concerned with determining if two designs are isomorphic. Here the v-set is any group G and translation by any element in G gives a design automorphism. Given a design B and its difference family D, two topological spaces, B and D, are constructed. We give topological conditions which imply that a design isomorphism is a group isomorphism.

Identiferoai:union.ndltd.org:unt.edu/info:ark/67531/metadc331968
Date12 1900
CreatorsSomporn Sutinuntopas
ContributorsBrand, Neal E., Kung, Joseph P. S., Bator, Elizabeth M., Vaughan, Nick H., Renka, Robert J.
PublisherUniversity of North Texas
Source SetsUniversity of North Texas
LanguageEnglish
Detected LanguageEnglish
TypeThesis or Dissertation
Formatiii, 78 leaves : ill., Text
RightsPublic, Somporn Sutinuntopas, Copyright, Copyright is held by the author, unless otherwise noted. All rights reserved.

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