scholarly journals Co-degrees Resilience for Perfect Matchings in Random Hypergraphs

10.37236/8167 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Asaf Ferber ◽  
Lior Hirschfeld
Keyword(s):  
High Probability ◽  
Perfect Matching ◽  
Perfect Matchings ◽  
Uniform Hypergraph ◽  
Hypergraph Model ◽  
Large N ◽  
Random Hypergraphs

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. That is, for a sufficiently large $n$ which is divisible by $k$, and $p\geq C_k\log {n}/n$, we prove that with high probability every subgraph $H\subseteq H^k_{n,p}$ with minimum co-degree (meaning, the number of supersets every set of size $k-1$ is contained in) at least $(1/2+o(1))np$ contains a perfect matching.

Rainbow Perfect Matchings in Complete Bipartite Graphs: Existence and Counting

2013 ◽  
Vol 22 (5) ◽  
pp. 783-799 ◽  
Author(s):  
GUILLEM PERARNAU ◽  
ORIOL SERRA
Keyword(s):  
Bipartite Graph ◽  
High Probability ◽  
Perfect Matching ◽  
Complete Bipartite Graph ◽  
Bipartite Graphs ◽  
Perfect Matchings ◽  
Asymptotic Enumeration ◽  
Complete Bipartite Graphs ◽  
Edge Colourings ◽  
Square Matrix

A perfect matchingMin an edge-coloured complete bipartite graphKn,nis rainbow if no pair of edges inMhave the same colour. We obtain asymptotic enumeration results for the number of rainbow perfect matchings in terms of the maximum number of occurrences of each colour. We also consider two natural models of random edge-colourings ofKn,nand show that if the number of colours is at leastn, then there is with high probability a rainbow perfect matching. This in particular shows that almost every square matrix of ordernin which every entry appearsntimes has a Latin transversal.

scholarly journals Threshold and Hitting Time for High-Order Connectedness in Random Hypergraphs

10.37236/5064 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Oliver Cooley ◽  
Mihyun Kang ◽  
Christoph Koch
Keyword(s):  
High Probability ◽  
Hitting Time ◽  
Uniform Hypergraph ◽  
Uniform Hypergraphs ◽  
Time Result ◽  
Edge Probability ◽  
The Moment ◽  
Definition Of ◽  
Random Hypergraph ◽  
Random Hypergraphs

We consider the following definition of connectedness in $k$-uniform hypergraphs: two $j$-sets (sets of $j$ vertices) are $j$-connected if there is a walk of edges between them such that two consecutive edges intersect in at least $j$ vertices. The hypergraph is $j$-connected if all $j$-sets are pairwise $j$-connected. We determine the threshold at which the random $k$-uniform hypergraph with edge probability $p$ becomes $j$-connected with high probability. We also deduce a hitting time result for the random hypergraph process – the hypergraph becomes $j$-connected at exactly the moment when the last isolated $j$-set disappears. This generalises the classical hitting time result of Bollobás and Thomason for graphs.

scholarly journals Perfect Matchings in Random r-regular, s-uniform Hypergraphs

1996 ◽  
Vol 5 (1) ◽  
pp. 1-14 ◽  
Author(s):  
Colin Cooper ◽  
Alan Frieze ◽  
Michael Molloy ◽  
Bruce Reed
Keyword(s):  
High Probability ◽  
Perfect Matching ◽  
Perfect Matchings ◽  
Expected Number ◽  
Uniform Hypergraphs ◽  
The Difference

We show that r-regular, s-uniform hypergraphs contain a perfect matching with high probability (whp), provided The Proof is based on the application of a technique of Robinson and Wormald [7, 8]. The space of hypergraphs is partitioned into subsets according to the number of small cycles in the hypergraph. The difference in the expected number of perfect matchings between these subsets explains most of the variance of the number of perfect matchings in the space of hypergraphs, and is sufficient to prove existence (whp), using the Chebychev Inequality.

scholarly journals The Size of the Giant Component in Random Hypergraphs: a Short Proof

10.37236/7712 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Oliver Cooley ◽  
Mihyun Kang ◽  
Christoph Koch
Keyword(s):  
High Probability ◽  
Short Proof ◽  
The Other ◽  
Connected Components ◽  
Search Process ◽  
Uniform Hypergraph ◽  
Breadth First Search ◽  
Asymptotic Size ◽  
Uniform Hypergraphs ◽  
Random Hypergraphs

We consider connected components in $k$-uniform hypergraphs for the following notion of connectedness: given integers $k\ge 2$ and $1\le j \le k-1$, two $j$-sets (of vertices) lie in the same $j$-component if there is a sequence of edges from one to the other such that consecutive edges intersect in at least $j$ vertices.We prove that certain collections of $j$-sets constructed during a breadth-first search process on $j$-components in a random $k$-uniform hypergraph are reasonably regularly distributed with high probability. We use this property to provide a short proof of the asymptotic size of the giant $j$-component shortly after it appears.

scholarly journals Tight Co-Degree Condition for Perfect Matchings in 4-Graphs

10.37236/2338 ◽  
2012 ◽  
Vol 19 (2) ◽  
Author(s):  
Andrzej Czygrinow ◽  
Vikram Kamat
Keyword(s):  
Perfect Matching ◽  
Perfect Matchings ◽  
Uniform Hypergraph ◽  
Degree Condition

We will give a tight minimum co-degree condition for a 4-uniform hypergraph to contain a perfect matching.

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 On the Number of Perfect Matchings in Random Lifts

2010 ◽  
Vol 19 (5-6) ◽  
pp. 791-817 ◽  
Author(s):  
CATHERINE GREENHILL ◽  
SVANTE JANSON ◽  
ANDRZEJ RUCIŃSKI
Keyword(s):  
Asymptotic Formula ◽  
Complete Graph ◽  
Pairwise Disjoint ◽  
Perfect Matching ◽  
Perfect Matchings ◽  
Lattice Points ◽  
Laplace’S Method ◽  
Second Moment ◽  
Multiple Dimensions ◽  
Laplace's Method

Let G be a fixed connected multigraph with no loops. A random n-lift of G is obtained by replacing each vertex of G by a set of n vertices (where these sets are pairwise disjoint) and replacing each edge by a randomly chosen perfect matching between the n-sets corresponding to the endpoints of the edge. Let XG be the number of perfect matchings in a random lift of G. We study the distribution of XG in the limit as n tends to infinity, using the small subgraph conditioning method.We present several results including an asymptotic formula for the expectation of XG when G is d-regular, d ≥ 3. The interaction of perfect matchings with short cycles in random lifts of regular multigraphs is also analysed. Partial calculations are performed for the second moment of XG, with full details given for two example multigraphs, including the complete graph K4.To assist in our calculations we provide a theorem for estimating a summation over multiple dimensions using Laplace's method. This result is phrased as a summation over lattice points, and may prove useful in future applications.

scholarly journals The Multiple-Orientability Thresholds for Random Hypergraphs

2015 ◽  
Vol 25 (6) ◽  
pp. 870-908 ◽  
Author(s):  
NIKOLAOS FOUNTOULAKIS ◽  
MEGHA KHOSLA ◽  
KONSTANTINOS PANAGIOTOU
Keyword(s):  
Load Balancing ◽  
Hard Disk ◽  
Maximum Load ◽  
Disk Arrays ◽  
Uniform Hypergraph ◽  
Cuckoo Hashing ◽  
Uniform Hypergraphs ◽  
Random Hypergraphs ◽  
Do So ◽  
Parallel Access

Ak-uniform hypergraphH= (V, E) is called ℓ-orientable if there is an assignment of each edgee∈Eto one of its verticesv∈esuch that no vertex is assigned more than ℓ edges. LetHn,m,kbe a hypergraph, drawn uniformly at random from the set of allk-uniform hypergraphs withnvertices andmedges. In this paper we establish the threshold for the ℓ-orientability ofHn,m,kfor allk⩾ 3 and ℓ ⩾ 2, that is, we determine a critical quantityc*k,ℓsuch that with probability 1 −o(1) the graphHn,cn,khas an ℓ-orientation ifc<c*k,ℓ, but fails to do so ifc>c*k,ℓ.Our result has various applications, including sharp load thresholds for cuckoo hashing, load balancing with guaranteed maximum load, and massive parallel access to hard disk arrays.

scholarly journals Face-Width of Pfaffian Braces and Polyhex Graphs on Surfaces

10.37236/3540 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Dong Ye ◽  
Heping Zhang
Keyword(s):  
Bipartite Graph ◽  
Polynomial Time ◽  
Perfect Matching ◽  
Cartesian Product ◽  
Perfect Matchings ◽  
Face Width ◽  
Graphs On Surfaces ◽  
Positive Genus ◽  
Pfaffian Graph

A graph $G$ with a perfect matching is Pfaffian if it admits an orientation $D$ such that every central cycle $C$ (i.e. $C$ is of even size and $G-V(C)$ has a perfect matching) has an odd number of edges oriented in either direction of the cycle. It is known that the number of perfect matchings of a Pfaffian graph can be computed in polynomial time. In this paper, we show that every embedding of a Pfaffian brace (i.e. 2-extendable bipartite graph)  on a surface with a positive genus has face-width at most 3.  Further, we study Pfaffian cubic braces and obtain a characterization of Pfaffian polyhex graphs: a polyhex graph is Pfaffian if and only if it is either non-bipartite or isomorphic to the cube, or the Heawood graph, or the Cartesian product $C_k\times K_2$ for even integers $k\ge 6$.

scholarly journals Commuting decomposition of Kn1,n2,...,nk through realization of the product A(G)A(GPk )

2018 ◽  
Vol 6 (1) ◽  
pp. 343-356
Author(s):  
K. Arathi Bhat ◽  
G. Sudhakara
Keyword(s):  
Chromatic Number ◽  
Perfect Matching ◽  
Domination Number ◽  
Perfect Matchings ◽  
Factor Graph ◽  
Vertex Set ◽  
Graph Parameters

Abstract In this paper, we introduce the notion of perfect matching property for a k-partition of vertex set of given graph. We consider nontrivial graphs G and GPk , the k-complement of graph G with respect to a kpartition of V(G), to prove that A(G)A(GPk ) is realizable as a graph if and only if P satis_es perfect matching property. For A(G)A(GPk ) = A(Γ) for some graph Γ, we obtain graph parameters such as chromatic number, domination number etc., for those graphs and characterization of P is given for which GPk and Γ are isomorphic. Given a 1-factor graph G with 2n vertices, we propose a partition P for which GPk is a graph of rank r and A(G)A(GPk ) is graphical, where n ≤ r ≤ 2n. Motivated by the result of characterizing decomposable Kn,n into commuting perfect matchings [2], we characterize complete k-partite graph Kn1,n2,...,nk which has a commuting decomposition into a perfect matching and its k-complement.

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