Title:
Matching problems in hypergraphs
Matching problems in hypergraphs
| dc.contributor.advisor | Yu, Xingxing | |
| dc.contributor.author | Yuan, Xiaofan | |
| dc.contributor.committeeMember | Bernshteyn, Anton | |
| dc.contributor.committeeMember | Huang, Hao | |
| dc.contributor.committeeMember | Vempala, Santosh | |
| dc.contributor.committeeMember | Yu, Josephine | |
| dc.contributor.department | Mathematics | |
| dc.date.accessioned | 2022-08-25T13:38:19Z | |
| dc.date.available | 2022-08-25T13:38:19Z | |
| dc.date.created | 2022-08 | |
| dc.date.issued | 2022-07-30 | |
| dc.date.submitted | August 2022 | |
| dc.date.updated | 2022-08-25T13:38:19Z | |
| dc.description.abstract | Kühn, Osthus, and Treglown and, independently, Khan proved that if H is a 3-uniform hypergraph on n vertices, where n is a multiple of 3 and large, and the minimum vertex degree of H is greater than {(n-1) choose 2} - {2n/3 choose 2}, then H contains a perfect matching. We show that for sufficiently large n divisible by 3, if F_1, ..., F_{n/3} are 3-uniform hypergraphs with a common vertex set and the minimum vertex degree in each F_i is greater than {(n-1) choose 2} - {2n/3 choose 2} for i = 1, ..., n/3, then the family {F_1, ..., F_{n/3}} admits a rainbow matching, i.e., a matching consisting of one edge from each F_i. This is done by converting the rainbow matching problem to a perfect matching problem in a special class of uniform hypergraphs. We also prove that, for any integers k, l with k >= 3 and k/2 < l <= k-1, there exists a positive real μ such that, for all sufficiently large integers m, n satisfying n/k - μn <= m <= n/k - 1 - (1 - l/k){ceil of (k - l)/(2l - k)}, if H is a k-uniform hypergraph on n vertices and the minimum l-degree of H is greater than {(n-l) choose (k-l)} - {(n-l-m) choose (k-l)}, then H has a matching of size m+1. This improves upon an earlier result of Hàn, Person, and Schacht for the range k/2 < l <= k-1. In many cases, our result gives tight bound on the minimum l-degree of H for near perfect matchings. For example, when l >= 2k/3, n ≡ r (mod k), 0 <= r < k, and r + l >= k, we can take m to be the minimum integer at least n/k - 2. | |
| dc.description.degree | Ph.D. | |
| dc.format.mimetype | application/pdf | |
| dc.identifier.uri | http://hdl.handle.net/1853/67294 | |
| dc.language.iso | en_US | |
| dc.publisher | Georgia Institute of Technology | |
| dc.subject | Perfect matching | |
| dc.subject | Near perfect matching | |
| dc.subject | Rainbow matching | |
| dc.subject | Fractional matching | |
| dc.subject | Hypergraph | |
| dc.title | Matching problems in hypergraphs | |
| dc.type | Text | |
| dc.type.genre | Dissertation | |
| dspace.entity.type | Publication | |
| local.contributor.advisor | Yu, Xingxing | |
| local.contributor.corporatename | College of Sciences | |
| local.contributor.corporatename | School of Mathematics | |
| relation.isAdvisorOfPublication | 3b32a3b5-5417-4c47-8a35-79346368e87f | |
| relation.isOrgUnitOfPublication | 85042be6-2d68-4e07-b384-e1f908fae48a | |
| relation.isOrgUnitOfPublication | 84e5d930-8c17-4e24-96cc-63f5ab63da69 | |
| thesis.degree.level | Doctoral |