Packing perfect matchings in random hypergraphs

Ferber, Asaf; Vu, Van

doi:10.1002/rsa.20745

Cited by 3 publications

(10 citation statements)

References 19 publications

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“…Following an idea introduced in Ferber and Vu, in this stage we give a randomized algorithm which w.v.h.p. finds edge disjoint copies of P 1 ,…, P t in D 1 ∼

scriptD

( n, p 1 ).…”

Section: Packing Arbitrarily Oriented Hamilton Cycles In Scriptd(n P)mentioning

confidence: 99%

“…We will gradually expose D 1 using the coins

{false{C_{i}^{e} false}}_{i \in false[t false]}

, always maintaining a "fresh coin” for each unused edge. Provided we never examine more than the first M coins for any e ∈ E ( D n ), the above coupling shows that the exposed random digraph is generated according to ( n, p 1 ) (for more details about this idea, the reader is referred to Ferber and Vu).…”

Section: Packing Arbitrarily Oriented Hamilton Cycles In Scriptd(n P)mentioning

confidence: 99%

“…Enhancing a recent "online sprinkling" technique introduced by Ferber and Vu, we manage to tackle these two problems. Our first theorem gives an asymptotically optimal result for packing arbitrarily oriented Hamilton cycles in

scriptD

∼

scriptD

( n, p ).…”

Section: Introductionmentioning

confidence: 99%

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Packing and counting arbitrary Hamilton cycles in random digraphs

Ferber

Long

2018

Random Struct Algorithms

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We prove packing and counting theorems for arbitrarily oriented Hamilton cycles in scriptD(n, p) for nearly optimal p (up to a logcn factor). In particular, we show that given t = (1 − o(1))np Hamilton cycles C1,…,Ct, each of which is oriented arbitrarily, a digraph scriptD∼scriptD(n, p) w.h.p. contains edge disjoint copies of C1,…,Ct, provided p=ωfalse(log3nfalse/nfalse). We also show that given an arbitrarily oriented n‐vertex cycle C, a random digraph scriptD∼scriptD(n, p) w.h.p. contains (1 ± o(1))n!pn copies of C, provided p≥log1+ofalse(1false)nfalse/n.

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“…Following an idea introduced in Ferber and Vu, in this stage we give a randomized algorithm which w.v.h.p. finds edge disjoint copies of P 1 ,…, P t in D 1 ∼

scriptD

( n, p 1 ).…”

Section: Packing Arbitrarily Oriented Hamilton Cycles In Scriptd(n P)mentioning

confidence: 99%

“…We will gradually expose D 1 using the coins

{false{C_{i}^{e} false}}_{i \in false[t false]}

Section: Packing Arbitrarily Oriented Hamilton Cycles In Scriptd(n P)mentioning

confidence: 99%

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Packing and counting arbitrary Hamilton cycles in random digraphs

Ferber

Long

2018

Random Struct Algorithms

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“…The proof of Theorem utilises and extends the ‘online sprinkling’ technique introduced by the first author and Vu . Roughly speaking, we embed our trees and expose

G_{n, p}

together, edge by edge, making sure that the trees are embedded disjointly and each discovered edge of

G_{n, p}

is used in the embedding.…”

Section: Introductionmentioning

confidence: 99%

Packing trees of unbounded degrees in random graphs

Ferber

Samotij

2018

J. London Math. Soc.

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In this paper, we address the problem of packing large trees in Gn,p. In particular, we prove the following result. Suppose that T1,…,TN are n‐vertex trees, each of which has maximum degree at most (np)1/6/(logn)6. Then with high probability, one can find edge‐disjoint copies of all the Ti in the random graph Gn,p, provided that p⩾(logn)36/n and N⩽(1−ε)np/2 for a positive constant ε. Moreover, if each Ti has at most (1−α)n vertices, for some positive α, then the same result holds under the much weaker assumptions that p⩾(logn)2/false(cnfalse) and Δ(Ti)⩽cnp/logn for some c that depends only on α and ε. Our assumptions on maximum degrees of the trees are significantly weaker than those in all previously known approximate packing results.

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“…However, in the hypergraph setting, completely new techniques are often required. In this paper we utilize and modify a new approach due to Vu and the first author [5] to deal with the following problem (we discuss the precise definition of Hamilton cycles in hypergraphs below).…”

Section: Introductionmentioning

confidence: 99%

Packing Loose Hamilton Cycles

Ferber¹,

Luh²,

Montealegre³

et al. 2017

Combinator. Probab. Comp.

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Abstract. A subset C of edges in a k-uniform hypergraph H is a loose Hamilton cycle if C covers all the vertices of H and there exists a cyclic ordering of these vertices such that the edges in C are segments of that order and such that every two consecutive edges share exactly one vertex. The binomial random k-uniform hypergraph H k n,p has vertex set [n] and an edge set E obtained by adding each k-tuple e ∈[n] k to E with probability p, independently at random. Here we consider the problem of finding edge-disjoint loose Hamilton cycles covering all but o(|E|) edges, referred to as the packing problem. While it is known that the threshold probability for the appearance of a loose Hamilton cycle in H k n,p is p = Θ log n n k−1 , the best known bounds for the packing problem are around p = polylog(n)/n. Here we make substantial progress and prove the following asymptotically (up to a polylog(n) factor) best possible result: For p ≥ log C n/n k−1 , a random k-uniform hypergraph H k n,p with high probability contains N :edgedisjoint loose Hamilton cycles.Our proof utilizes and modifies the idea of "online sprinkling" recently introduced by Vu and the first author.

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Packing perfect matchings in random hypergraphs

Cited by 3 publications

References 19 publications

Packing and counting arbitrary Hamilton cycles in random digraphs

Packing and counting arbitrary Hamilton cycles in random digraphs

Packing trees of unbounded degrees in random graphs

Packing Loose Hamilton Cycles

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