Matching problems in hypergraphs

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.

Level thesis:degree_level

Doctoral

Department dc:contributor.department

Mathematics

Grantor dc:publisher

Georgia Institute of Technology

Year dc:date.issued

2022

Author dc:creator

• Yuan, Xiaofan

Advisor dc:contributor.advisor

• Yu, Xingxing

Committee members dc:contributor.committeemember

• Bernshteyn, Anton
• Huang, Hao
• Vempala, Santosh
• Yu, Josephine

• Perfect matching
• Near perfect matching
• Rainbow matching
• Fractional matching
• Hypergraph

Language dc:language.iso

en_US

Handle dc:identifier.uri

http://hdl.handle.net/1853/67294

OAI identifier oai:identifier

oai:repository.gatech.edu:1853/67294