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Xiaofan Yuan
(Advisor: Prof. Xingxing Yu) will defend a doctoral thesis entitled,
Matching Problems in Hypergraphs
On
Thursday, June 9 at 10:00 a.m.
Skiles 006
https://gatech.zoom.us/j/91659544858?pwd=SWZtVG15dGFiWEFXSHR1U0JNbVVBZz09
Abstract
Ku¨hn, Osthus, and Treglown and, independently, Khan proved that if H is a 3-uniform hypergraph
with n vertices, where n ∈ 3Z and large, and δ₁(H) > ⁿ−¹ − 2n/3 , then H contains a perfect
We show that for n 3Z sufficiently large, if F , . . . , F are 3-uniform hypergraphs
with a common vertex set and δ₁(Fi) > (n−1)−(2n/3) for i ∈ [n/3], then {F₁, . . . , Fn/3} admits a
rainbow
matching, i.e., a matching consisting of one edge from each Fi. 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
l
l
2l − k
,
if H is a k-uniform hypergraph on n vertices and δl(H) >
(n−
) −
) hen H has a
. This improves upon an earlier result of H`an, Person, and Schacht for the range k/2 < l ≤ k − 1.
In many cases, our result gives tight bound on δl(H) for near perfect matchings (e.g., when l ≥
2k/3, n ≡ r (mod k), 0 ≤ r < k, and r + l ≥ k, we can take m = n/k − 2).
Committee
• Prof. Anton Bernshteyn - School of Mathematics, Georgia Institute of Technology
• Prof. Hao Huang - Department of Mathematics, National University of Singapore (reader)
• Prof. Santosh Vempala - College of Computing, Georgia Institute of Technology
• Prof. Josephine Yu - School of Mathematics, Georgia Institute of Technology
• Prof. Xingxing Yu - School of Mathematics, Georgia Institute of Technology (advisor)
