On Perfect Matchings in Uniform Hypergraphs with Large Minimum Vertex Degree

2009 ◽  
Vol 23 (2) ◽  
pp. 732-748 ◽  
Author(s):  
Hip Hàn ◽  
Yury Person ◽  
Mathias Schacht
Keyword(s):  
Perfect Matchings ◽  
Vertex Degree ◽  
Uniform Hypergraphs

scholarly journals Vertex Degree Sums for Perfect Matchings in 3-Uniform Hypergraphs

10.37236/7658 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Yi Zhang ◽  
Yi Zhao ◽  
Mei Lu
Keyword(s):  
Perfect Matching ◽  
Minimum Degree ◽  
Perfect Matchings ◽  
Vertex Degree ◽  
Uniform Hypergraph ◽  
Degree Sum ◽  
Uniform Hypergraphs ◽  
Isolated Vertex ◽  
The One

We determine the minimum degree sum of two adjacent vertices that ensures a perfect matching in a 3-uniform hypergraph without an isolated vertex. Suppose that $H$ is a 3-uniform hypergraph whose order $n$ is sufficiently large and divisible by $3$. If $H$ contains no isolated vertex and $\deg(u)+\deg(v) > \frac{2}{3}n^2-\frac{8}{3}n+2$ for any two vertices $u$ and $v$ that are contained in some edge of $H$, then $H$ contains a perfect matching. This bound is tight and the (unique) extremal hyergraph is a different space barrier from the one for the corresponding Dirac problem.

Nearly Perfect Matchings in Uniform Hypergraphs

2021 ◽  
Vol 35 (2) ◽  
pp. 1022-1049
Author(s):  
Hongliang Lu ◽  
Xingxing Yu ◽  
Xiaofan Yuan
Keyword(s):  
Perfect Matchings ◽  
Uniform Hypergraphs

scholarly journals Perfect matchings in large uniform hypergraphs with large minimum collective degree

2009 ◽  
Vol 116 (3) ◽  
pp. 613-636 ◽  
Author(s):  
Vojtech Rödl ◽  
Andrzej Ruciński ◽  
Endre Szemerédi
Keyword(s):  
Perfect Matchings ◽  
Uniform Hypergraphs ◽  
Collective Degree

scholarly journals Triangle-degrees in graphs and tetrahedron coverings in 3-graphs

2020 ◽  
pp. 1-25
Author(s):  
Victor Falgas-Ravry ◽  
Klas Markström ◽  
Yi Zhao
Keyword(s):  
Upper Bound ◽  
Vertex Degree ◽  
Covering Problem ◽  
Uniform Hypergraphs ◽  
Vertex Graph ◽  
Optimal Bounds ◽  
Special Instance ◽  
Tripartite Graphs

Abstract We investigate a covering problem in 3-uniform hypergraphs (3-graphs): Given a 3-graph F, what is c1(n, F), the least integer d such that if G is an n-vertex 3-graph with minimum vertex-degree $\delta_1(G)>d$ then every vertex of G is contained in a copy of F in G? We asymptotically determine c1(n, F) when F is the generalized triangle $K_4^{(3)-}$ , and we give close to optimal bounds in the case where F is the tetrahedron $K_4^{(3)}$ (the complete 3-graph on 4 vertices). This latter problem turns out to be a special instance of the following problem for graphs: Given an n-vertex graph G with $m> n^2/4$ edges, what is the largest t such that some vertex in G must be contained in t triangles? We give upper bound constructions for this problem that we conjecture are asymptotically tight. We prove our conjecture for tripartite graphs, and use flag algebra computations to give some evidence of its truth in the general case.

scholarly journals Near Perfect Matchings ink-Uniform Hypergraphs

2014 ◽  
Vol 24 (5) ◽  
pp. 723-732 ◽  
Author(s):  
JIE HAN
Keyword(s):  
Perfect Matchings ◽  
Large Integer ◽  
Uniform Hypergraph ◽  
Uniform Hypergraphs

LetHbe ak-uniform hypergraph onnvertices wherenis a sufficiently large integer not divisible byk. We prove that if the minimum (k− 1)-degree ofHis at least ⌊n/k⌋, thenHcontains a matching with ⌊n/k⌋ edges. This confirms a conjecture of Rödl, Ruciński and Szemerédi [13], who proved that minimum (k− 1)-degreen/k+O(logn) suffices. More generally, we show thatHcontains a matching of sizedif its minimum codegree isd<n/k, which is also best possible.

scholarly journals Vertex Degree Sums for Matchings in 3-Uniform Hypergraphs

10.37236/8627 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Yi Zhang ◽  
Yi Zhao ◽  
Mei Lu
Keyword(s):  
Vertex Degree ◽  
Uniform Hypergraph ◽  
Degree Sum ◽  
Uniform Hypergraphs ◽  
Positive Integers ◽  
Degree Sum Condition

Let $n, s$ be positive integers such that $n$ is sufficiently large and $s\le n/3$. Suppose $H$ is a 3-uniform hypergraph of order $n$ without isolated vertices. If $\deg(u)+\deg(v) > 2(s-1)(n-1)$ for any two vertices $u$ and $v$ that are contained in some edge of $H$, then $H$ contains a matching of size $s$. This degree sum condition is best possible and confirms a conjecture of the authors [Electron. J. Combin. 25 (3), 2018], who proved the case when $s= n/3$.

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