Tight minimum degree conditions forcing perfect matchings in uniform hypergraphs

2013 ◽  
pp. 33-38
Author(s):  
Andrew Treglown ◽  
Yi Zhao
Keyword(s):  
Minimum Degree ◽  
Perfect Matchings ◽  
Uniform Hypergraphs ◽  
Degree Conditions

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.

scholarly journals Berge Cycles in Non-Uniform Hypergraphs

10.37236/9346 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Zoltán Füredi ◽  
Alexandr Kostochka ◽  
Ruth Luo
Keyword(s):  
Upper Bound ◽  
Minimum Degree ◽  
Extremal Problems ◽  
Dirac Type ◽  
Set Systems ◽  
Uniform Hypergraphs ◽  
Degree Conditions

We consider two extremal problems for set systems without long Berge cycles. First we give Dirac-type minimum degree conditions that force long Berge cycles. Next we give an upper bound for the number of hyperedges in a hypergraph with bounded circumference. Both results are best possible in infinitely many cases.

scholarly journals Perfect matchings in uniform hypergraphs with large minimum degree

2006 ◽  
Vol 27 (8) ◽  
pp. 1333-1349 ◽  
Author(s):  
Vojtech Rödl ◽  
Andrzej Ruciński ◽  
Endre Szemerédi
Keyword(s):  
Minimum Degree ◽  
Perfect Matchings ◽  
Uniform Hypergraphs

scholarly journals On Directed Versions of the Hajnal–Szemerédi Theorem

2015 ◽  
Vol 24 (6) ◽  
pp. 873-928 ◽  
Author(s):  
ANDREW TREGLOWN
Keyword(s):  
Directed Graph ◽  
Minimum Degree ◽  
Directed Graphs ◽  
Perfect Matchings ◽  
Large Order ◽  
Vertex Disjoint ◽  
Removal Lemma

We say that a (di)graph G has a perfect H-packing if there exists a set of vertex-disjoint copies of H which cover all the vertices in G. The seminal Hajnal–Szemerédi theorem characterizes the minimum degree that ensures a graph G contains a perfect Kr-packing. In this paper we prove the following analogue for directed graphs: Suppose that T is a tournament on r vertices and G is a digraph of sufficiently large order n where r divides n. If G has minimum in- and outdegree at least (1−1/r)n then G contains a perfect T-packing.In the case when T is a cyclic triangle, this result verifies a recent conjecture of Czygrinow, Kierstead and Molla [4] (for large digraphs). Furthermore, in the case when T is transitive we conjecture that it suffices for every vertex in G to have sufficiently large indegree or outdegree. We prove this conjecture for transitive triangles and asymptotically for all r ⩾ 3. Our approach makes use of a result of Keevash and Mycroft [10] concerning almost perfect matchings in hypergraphs as well as the Directed Graph Removal Lemma [1, 6].

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 Semi-Degree Threshold for Anti-Directed Hamiltonian Cycles

10.37236/3610 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Louis DeBiasio ◽  
Theodore Molla
Keyword(s):  
Directed Graph ◽  
Hamiltonian Cycle ◽  
Minimum Degree ◽  
Directed Graphs ◽  
Hamiltonian Cycles ◽  
Alternate Direction ◽  
Degree Conditions

In 1960 Ghouila-Houri extended Dirac's theorem to directed graphs by proving that if $D$ is a directed graph on $n$ vertices with minimum out-degree and in-degree at least $n/2$, then $D$ contains a directed Hamiltonian cycle. For directed graphs one may ask for other orientations of a Hamiltonian cycle and in 1980 Grant initiated the problem of determining minimum degree conditions for a directed graph $D$ to contain an anti-directed Hamiltonian cycle (an orientation in which consecutive edges alternate direction). We prove that for sufficiently large even $n$, if $D$ is a directed graph on $n$ vertices with minimum out-degree and in-degree at least $\frac{n}{2}+1$, then $D$ contains an anti-directed Hamiltonian cycle. In fact, we prove the stronger result that $\frac{n}{2}$ is sufficient unless $D$ is one of two counterexamples. This result is sharp.

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