#### Date of Award

8-12-2016

#### Degree Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics and Statistics

#### First Advisor

Yi Zhao

#### Abstract

We consider two extremal problems in hypergraphs. First, given k ≥ 3 and k-partite k-uniform hypergraphs, as a generalization of graph (k = 2) matchings, we determine the partite minimum codegree threshold for matchings with at most one vertex left in each part, thereby answering a problem asked by R ̈odl and Rucin ́ski. We further improve the partite minimum codegree conditions to sum of all k partite codegrees, in which case the partite minimum codegree is not necessary large.

Second, as a generalization of (hyper)graph matchings, we determine the minimum vertex degree threshold asymptotically for perfect K_{a,b,c}-tlings in large 3-uniform hypergraphs, where K_{a,b,c} is any complete 3-partite 3-uniform hypergraphs with each part of size a, b and c. This partially answers a question of Mycroft, who proved an analogous result with respect to codegree for r-uniform hypergraphs for all r ≥ 3. Our proof uses Regularity Lemma, the absorbing method, fractional tiling, and a recent result on shadows for 3-graphs.

#### Recommended Citation

Zang, Chuanyun, "Matchings and Tilings in Hypergraphs." Dissertation, Georgia State University, 2016.

http://scholarworks.gsu.edu/math_diss/31