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### Perfect matchings in random uniform hypergraphs

Jeong Han Kim
2003 Random structures & algorithms (Print)
In the random k-uniform hypergraph H k (n, p) on a vertex set V of size n, each subset of size k of V independently belongs to it with probability p.  ...  Motivated by a theorem of Erdős and Rényi [6] regarding when a random graph G(n, p) = H 2 (n, p) has a perfect matching, Schmidt and Shamir [14] essentially conjectured the following.  ...  Molloy and Reed [5] found the exact constant c k (for a fixed k) so that with high probability the random d-regular k-uniform hypergraph has a perfect matching if d ≥ c k , otherwise no perfect matching  ...

### Almost perfect matchings in random uniform hypergraphs

Michael Krivelevich
1997 Discrete Mathematics
A random r-uniform hypergraph ~r(n,p) is an r-uniform hypergraph with vertex set V of size IV] =n, in which each r-subset of V is chosen to be an edge of H E orgy(n, p) with probability p (where p may  ...  One of the central problems in probabilistic combinatorics is that of determining the minimal probability p= p(n), for which a random hypergraph H c ~(n, p) has whpla perfect matching (assuming of course  ...  matching in an r-uniform hypergraph.  ...

### Perfect Matchings in Random r-regular, s-uniform Hypergraphs

Colin Cooper, Alan Frieze, Michael Molloy, Bruce Reed
1996 Combinatorics, probability & computing

### Perfect matchings in random s-uniform hypergraphs

Frieze, Svante Janson
2018
Cooper, Frieze, Molloy and Reed [5] considered the problem of perfect matchings in random r-regular, s-uniform hypergraphs.  ...  A set of edges M = {X{ : i € /} is a perfect matching if (i) ijLje.1 implies X{ n Xj• = 0, and In this paper we consider the question of whether a random s-uniform hypergraph contains a perfect matching  ...

### Perfect matchings in random r-regular, s-uniform hypergraphs

Colin Cooper
2018
Thus if F is simple it has a perfect matching if and only if 7(F) has a perfect matching.  ...  Let Q = Q(n,r,s) = {G = (V,E) : G is r-regular and 5-uniform }. Let G = G n ,r,s be chosen uniformly at random from Q.  ...

### Matchings and Hamilton cycles in hypergraphs

Daniela Kühn, Deryk Osthus
2005 Discrete Mathematics & Theoretical Computer Science
International audience It is well known that every bipartite graph with vertex classes of size \$n\$ whose minimum degree is at least \$n/2\$ contains a perfect matching.  ...  We prove an analogue of this result for uniform hypergraphs.  ...  This in turn uses a probabilistic argument based on results about random perfect matchings in pseudo-random graphs [KOc] .  ...

### Cycles and Matchings in Randomly Perturbed Digraphs and Hypergraphs

MICHAEL KRIVELEVICH, MATTHEW KWAN, BENNY SUDAKOV
2016 Combinatorics, probability & computing
First, we prove that adding linearly many random edges to a densek-uniform hypergraph ensures the (asymptotically almost sure) existence of a perfect matching or a loose Hamilton cycle.  ...  We give several results showing that different discrete structures typically gain certain spanning substructures (in particular, Hamilton cycles) after a modest random perturbation.  ...  In particular, a k-uniform tight cycle is a k-uniform hypergraph with a cyclic ordering on its vertices such that every k consecutive vertices form an edge.  ...

### Cycles and matchings in randomly perturbed digraphs and hypergraphs

Michael Krivelevich, Matthew Kwan, Benny Sudakov
2015 Electronic Notes in Discrete Mathematics
Our first theorem is that adding linearly many random edges to a dense k-uniform hypergraph typically ensures the existence of a perfect matching or a loose Hamilton cycle.  ...  In these situations we show that the extremal examples are "fragile" in that after a modest random perturbation our desired substructures will typically appear.  ...  In particular, a k-uniform tight cycle is a k-uniform hypergraph with a cyclic ordering on its vertices such that every k consecutive vertices form an edge.  ...

### Matchings in hypergraphs of large minimum degree

Daniela Kühn, Deryk Osthus
2006 Journal of Graph Theory
We also prove several related results which guarantee the existence of almost perfect matchings in r-uniform hypergraphs of large minimum degree.  ...  It is well known that every bipartite graph with vertex classes of size n whose minimum degree is at least n/2 contains a perfect matching. We prove an analogue of this result for hypergraphs.  ...  For random r-uniform hypergraphs, the threshold for a perfect matching is still not known. There are several partial results, see e.g. Kim [11] .  ...

### Co-degrees resilience for perfect matchings in random hypergraphs [article]

Asaf Ferber, Lior Hirschfeld
2019 arXiv   pre-print
In this paper we prove an optimal co-degrees resilience property for the binomial k-uniform hypergraph model H_n,p^k with respect to perfect matchings.  ...  ) at least (1/2+o(1))np contains a perfect matching.  ...  We are grateful to the referees for their valuable comments which were instrumental in revising this paper.  ...

### Co-degrees Resilience for Perfect Matchings in Random Hypergraphs

Asaf Ferber, Lior Hirschfeld
2020 Electronic Journal of Combinatorics
In this paper we prove an optimal co-degrees resilience property for the binomial \$k\$-uniform hypergraph model \$H_{n,p}^k\$ with respect to perfect matchings.  ...  ) at least \$(1/2+o(1))np\$ contains a perfect matching.  ...  We are grateful to the referees for their valuable comments which were instrumental in revising this paper.  ...

### Perfect matchings in uniform hypergraphs with large minimum degree

Vojtech Rödl, Andrzej Ruciński, Endre Szemerédi
2006 European journal of combinatorics (Print)
A perfect matching in a k-uniform hypergraph on n vertices, n divisible by k, is a set of n/k disjoint edges.  ...  We prove that for every k ≥ 3 and sufficiently large n, a perfect matching exists in every n-vertex k-uniform hypergraph in which each set of k − 1 vertices is contained in n/2 + Ω (log n) edges.  ...  In the case when r ≥ k − 2, Kühn and Osthus in [7] obtained an analogous result about almost perfect matchings in k-partite k-uniform hypergraphs.  ...

### Loose Hamilton Cycles in Random 3-Uniform Hypergraphs [article]

Alan Frieze
2010 arXiv   pre-print
In the random hypergraph H=H(n,p;3) each possible triple appears independently with probability p. A loose Hamilton cycle can be described as a sequence of edges x_i,y_i,x_i+1} for i=1,2,...,n/2.  ...  Let Γ = Γ(X, Y, p) be the random 3-uniform hypergraph where each triple in Ω is independently included with probability p.  ...  When ℓ = k − 1 we say that C is a loose Hamilton cycle and in this paper we will restrict our attention to loose Hamilton cycles in the random 3-uniform hypergraph H = H n,p;3 .  ...

### Distributed Algorithms for Matching in Hypergraphs [article]

Oussama Hanguir, Clifford Stein
2020 arXiv   pre-print
In this model, we present the first three parallel algorithms for d-Uniform Hypergraph Matching, and we analyse them in terms of resources such as memory usage, rounds of communication needed, and approximation  ...  We study the d-Uniform Hypergraph Matching (d-UHM) problem: given an n-vertex hypergraph G where every hyperedge is of size d, find a maximum cardinality set of disjoint hyperedges.  ...  The first contains random uniform hypergraphs, and the second contains random geometric hypergraphs. Random Uniform Hypergraphs.  ...

### Page 604 of Mathematical Reviews Vol. , Issue 93b [page]

1993 Mathematical Reviews
Intuitively, a “quasi-random” k-uniform hypergraph is one which has many of the properties associated with “almost all” k-uniform hypergraphs.  ...  K. (1-BELL6) Quasi-random classes of hypergraphs. Random Structures Algorithms 1 (1990), no. 4, 363-382.  ...
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