Disjoint union-free 3-uniform hypergraphs
| dc.contributor.author | Howard, Leah | |
| dc.contributor.supervisor | Dukes, Peter | |
| dc.date.accessioned | 2009-11-18T19:23:23Z | |
| dc.date.available | 2009-11-18T19:23:23Z | |
| dc.date.copyright | 2006 | en |
| dc.date.issued | 2006 | |
| dc.degree.department | Department of Mathematics and Statistics | |
| dc.degree.level | Master of Science M.Sc. | en |
| dc.description.abstract | A k-uniform hypergraph N = (X. B) of order n is a family of k-subsets B of an n-set X. A k-uniform hypergraph 7--L = (X. B) is disjoint union-free (DUF) if all disjoint pairs of elements of B have distinct unions; that is, if for every A, B, C, D E B. A fl B = C f1 D = 0 and A U B =CUD implies {A. B} = {C, D}. DUF families of maximum size have been studied by Erdos and Fiiredi. and in the case k = 3 this maximum size has been conjectured to equal (z). In this thesis, we study DUF 3-uniform hypergraphs with the main goals of presenting evidence to support this conjecture and studying the structures that have conjectured maximum size. If each pair of distinct elements of X is covered exactly A times in B then we call N = (X, B) an (n. k. A)-design. Using a blend of graph- and design-theoretic techniques, we study the DUF (n,. 3. 3)-designs that are the conjectured unique structures having maximum size. Central results of this thesis include substantially improving lower bounds on the maximum size for a large class of n. giving conditions on pair coverage in a DUF 3-uniform hypergraph that force an (n., 3, 3)-design, and providing constructions for DUF 3-uniform hypergraphs from families of DUF hypergraphs with smaller orders. Let. N = (X, B) be a DUF k-uniform hypergraph with the property that 7-t U {E} is not DUF for any k-subset E of X not already in H. Then N is maximally DUF. We introduce the problem of finding the minimum size of maximally DUF families and provide bounds on this quantity for k = 3. | en |
| dc.identifier.uri | http://hdl.handle.net/1828/1868 | |
| dc.language | English | eng |
| dc.language.iso | en | en |
| dc.rights | Available to the World Wide Web | en |
| dc.subject | hypergraphs | en |
| dc.subject.lcsh | UVic Subject Index::Sciences and Engineering::Mathematics::Mathematical statistics | en |
| dc.title | Disjoint union-free 3-uniform hypergraphs | en |
| dc.type | Thesis | en |