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 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 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 Saturation Number of Berge Stars in Random Hypergraphs

10.37236/9302 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
Lele Liu ◽  
Changxiang He ◽  
Liying Kang
Keyword(s):  
Uniform Hypergraph ◽  
Uniform Hypergraphs ◽  
Minimum Number ◽  
Saturation Number ◽  
Random Hypergraphs

Let $G$ be a graph. We say an $r$-uniform hypergraph $H$ is a Berge-$G$ if there exists a bijection $\phi: E(G)\to E(H)$ such that $e\subseteq\phi(e)$ for each $e\in E(G)$. Given a family of $r$-uniform hypergraphs $\mathcal{F}$ and an $r$-uniform hypergraph $H$, a spanning sub-hypergraph $H'$ of $H$ is $\mathcal{F}$-saturated in $H$ if $H'$ is $\mathcal{F}$-free, but adding any edge in $E(H)\backslash E(H')$ to $H'$ creates a copy of some $F\in\mathcal{F}$. The saturation number of $\mathcal{F}$ is the minimum number of edges in an $\mathcal{F}$-saturated spanning sub-hypergraph of $H$. In this paper, we asymptotically determine the saturation number of Berge stars in random $r$-uniform hypergraphs.

scholarly journals SYMMETRIC AND ASYMMETRIC RAMSEY PROPERTIES IN RANDOM HYPERGRAPHS

2017 ◽  
Vol 5 ◽  
Author(s):  
LUCA GUGELMANN ◽  
RAJKO NENADOV ◽  
YURY PERSON ◽  
NEMANJA ŠKORIĆ ◽  
ANGELIKA STEGER ◽  
...  
Keyword(s):  
Random Graph ◽  
Upper Bound ◽  
Natural Extension ◽  
Fixed Number ◽  
The Other ◽  
Ramsey Problem ◽  
Uniform Hypergraphs ◽  
General Bound ◽  
Random Hypergraphs ◽  
Monochromatic Copy

A celebrated result of Rödl and Ruciński states that for every graph $F$, which is not a forest of stars and paths of length 3, and fixed number of colours $r\geqslant 2$ there exist positive constants $c,C$ such that for $p\leqslant cn^{-1/m_{2}(F)}$ the probability that every colouring of the edges of the random graph $G(n,p)$ contains a monochromatic copy of $F$ is $o(1)$ (the ‘0-statement’), while for $p\geqslant Cn^{-1/m_{2}(F)}$ it is $1-o(1)$ (the ‘1-statement’). Here $m_{2}(F)$ denotes the 2-density of $F$. On the other hand, the case where $F$ is a forest of stars has a coarse threshold which is determined by the appearance of a certain small subgraph in $G(n,p)$. Recently, the natural extension of the 1-statement of this theorem to $k$-uniform hypergraphs was proved by Conlon and Gowers and, independently, by Friedgut, Rödl and Schacht. In particular, they showed an upper bound of order $n^{-1/m_{k}(F)}$ for the 1-statement, where $m_{k}(F)$ denotes the $k$-density of $F$. Similarly as in the graph case, it is known that the threshold for star-like hypergraphs is given by the appearance of small subgraphs. In this paper we show that another type of threshold exists if $k\geqslant 4$: there are $k$-uniform hypergraphs for which the threshold is determined by the asymmetric Ramsey problem in which a different hypergraph has to be avoided in each colour class. Along the way we obtain a general bound on the 1-statement for asymmetric Ramsey properties in random hypergraphs. This extends the work of Kohayakawa and Kreuter, and of Kohayakawa, Schacht and Spöhel who showed a similar result in the graph case. We prove the corresponding 0-statement for hypergraphs satisfying certain balancedness conditions.

scholarly journals Note on Upper Density of Quasi-Random Hypergraphs

10.37236/3222 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Vindya Bhat ◽  
Vojtěch Rödl
Keyword(s):  
Recent Result ◽  
Uniform Hypergraph ◽  
Random Sequences ◽  
Uniform Hypergraphs ◽  
Upper Density ◽  
Random Hypergraphs ◽  
Random Property

In 1964, Erdős proved that for any $\alpha > 0$, an $l$-uniform hypergraph $G$ with $n \geq n_0(\alpha, l)$ vertices and $\alpha \binom{n}{l}$ edges contains a large complete $l$-equipartite subgraph. This implies that any sufficiently large $G$ with density $\alpha > 0$ contains a large subgraph with density at least $l!/l^l$.In this note we study a similar problem for $l$-uniform hypergraphs $Q$ with a weak quasi-random property (i.e. with edges uniformly distributed over the sufficiently large subsets of vertices). We prove that any sufficiently large quasi-random $l$-uniform hypergraph $Q$ with density $\alpha > 0$ contains a large subgraph with density at least $\frac{(l-1)!}{l^{l-1}-1}$. In particular, for $l=3$, any sufficiently large such $Q$ contains a large subgraph with density at least $\frac{1}{4}$ which is the best possible lower bound.We define jumps for quasi-random sequences of $l$-graphs and our result implies that every number between 0 and $\frac{(l-1)!}{l^{l-1}-1}$ is a jump for quasi-random $l$-graphs. For $l=3$ this interval can be improved based on a recent result of Glebov, Král' and Volec. We prove that every number between [0, 0.3192) is a jump for quasi-random $3$-graphs.

scholarly journals Largest Components in Random Hypergraphs

2018 ◽  
Vol 27 (5) ◽  
pp. 741-762 ◽  
Author(s):  
OLIVER COOLEY ◽  
MIHYUN KANG ◽  
YURY PERSON
Keyword(s):  
Connected Components ◽  
Bounded Degree ◽  
Uniform Hypergraphs ◽  
Original Contribution ◽  
Common Notion ◽  
Formula Group ◽  
The Common ◽  
Edge Probability ◽  
Image Position ◽  
Random Hypergraphs

In this paper we considerj-tuple-connected components in randomk-uniform hypergraphs (thej-tuple-connectedness relation can be defined by letting twoj-sets be connected if they lie in a common edge and considering the transitive closure; the casej= 1 corresponds to the common notion of vertex-connectedness). We show that the existence of aj-tuple-connected component containing Θ(nj)j-sets undergoes a phase transition and show that the threshold occurs at edge probability$$\frac{(k-j)!}{\binom{k}{j}-1}n^{j-k}.$$Our proof extends the recent short proof for the graph case by Krivelevich and Sudakov, which makes use of a depth-first search to reveal the edges of a random graph.Our main original contribution is abounded degree lemma, which controls the structure of the component grown in the search process.

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.

scholarly journals Perfect Fractional Matchings in $k$-Out Hypergraphs

10.37236/6890 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
Pat Devlin ◽  
Jeff Kahn
Keyword(s):  
High Probability ◽  
Short Proof ◽  
Analogous Result ◽  
Stopping Time ◽  
Uniform Hypergraph ◽  
Time Result

Extending the notion of (random) $k$-out graphs, we consider when the $k$-out hypergraph is likely to have a perfect fractional matching. In particular, we show that for each $r$ there is a $k=k(r)$ such that the $k$-out $r$-uniform hypergraph on $n$ vertices has a perfect fractional matching with high probability (i.e., with probability tending to $1$ as $n\to\infty$) and prove an analogous result for $r$-uniform $r$-partite hypergraphs. This is based on a new notion of hypergraph expansion and the observation that sufficiently expansive hypergraphs admit perfect fractional matchings. As a further application, we give a short proof of a stopping-time result originally due to Krivelevich.

scholarly journals Hamiltonian Berge cycles in random hypergraphs

2020 ◽  
pp. 1-11
Author(s):  
Deepak Bal ◽  
Ross Berkowitz ◽  
Pat Devlin ◽  
Mathias Schacht
Keyword(s):  
Optimal Stopping ◽  
High Probability ◽  
Minimum Degree ◽  
Stopping Time ◽  
Threshold Probability ◽  
Optimal Stopping Time ◽  
Uniform Hypergraphs ◽  
Time Result ◽  
Random Hypergraphs

Abstract In this note we study the emergence of Hamiltonian Berge cycles in random r-uniform hypergraphs. For $r\geq 3$ we prove an optimal stopping time result that if edges are sequentially added to an initially empty r-graph, then as soon as the minimum degree is at least 2, the hypergraph with high probability has such a cycle. In particular, this determines the threshold probability for Berge Hamiltonicity of the Erdős–Rényi random r-graph, and we also show that the 2-out random r-graph with high probability has such a cycle. We obtain similar results for weak Berge cycles as well, thus resolving a conjecture of Poole.

scholarly journals Finding tight Hamilton cycles in random hypergraphs faster

2020 ◽  
pp. 1-19
Author(s):  
Peter Allen ◽  
Christoph Koch ◽  
Olaf Parczyk ◽  
Yury Person
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Hamilton Cycle ◽  
Uniform Hypergraph ◽  
Hamilton Cycles ◽  
Second Moment ◽  
Uniform Hypergraphs ◽  
Edge Probability ◽  
Random Hypergraphs

Abstract In an r-uniform hypergraph on n vertices, a tight Hamilton cycle consists of n edges such that there exists a cyclic ordering of the vertices where the edges correspond to consecutive segments of r vertices. We provide a first deterministic polynomial-time algorithm, which finds a.a.s. tight Hamilton cycles in random r-uniform hypergraphs with edge probability at least C log3n/n. Our result partially answers a question of Dudek and Frieze, who proved that tight Hamilton cycles exist already for p = ω(1/n) for r = 3 and p = (e + o(1))/n for $r \ge 4$ using a second moment argument. Moreover our algorithm is superior to previous results of Allen, Böttcher, Kohayakawa and Person, and Nenadov and Škorić, in various ways: the algorithm of Allen et al. is a randomized polynomial-time algorithm working for edge probabilities $p \ge {n^{ - 1 + \varepsilon}}$ , while the algorithm of Nenadov and Škorić is a randomized quasipolynomial-time algorithm working for edge probabilities $p \ge C\mathop {\log }\nolimits^8 n/n$ .

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