Tight Hamilton cycles in random hypergraphs

Allen, Peter; Böttcher, Julia; Kohayakawa, Yoshiharu; Person, Yury

doi:10.1002/rsa.20519

Cited by 20 publications

(76 citation statements)

References 24 publications

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“…We remark that our proof of Theorem 2 also yields a deterministic polynomial time algorithm for finding a copy of the kth power of the Hamilton cycle. The proof technique (see Section 2.2 for an overview) is partly inspired by the methods used in [2] (which have similarities to those of Kühn and Osthus [28]). …”

Section: Our Resultsmentioning

confidence: 99%

“…For the first task, we make use of the reservoir method developed in [2] (see also [28] for a similar method). In essence, the fundamental idea of this method is to ensure that P ′ contains a sufficiently big proportion of vertices which are free to be taken out of P ′ and used otherwise.…”

Section: 2mentioning

confidence: 99%

“…Finally, by considering the choices only during the use of Lemma 19 with j = 0, we can estimate the number of kth powers of cycles which we construct are at least n−1000kn/(log n) 2 t=n/(200k log n) 2 1 − ν 2k k p k t ≥ n−1000kn/(log n) 2 t=n/(200k log n) 2 1 − ν 2 p k t ≥ 1 − ν 2 n p kn n!n −1001kn/(log n) 2 .…”

Section: Enumerating Powers Of Hamilton Cyclesmentioning

confidence: 99%

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Powers of Hamilton cycles in pseudorandom graphs

et al. 2016

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Abstract. We study the appearance of powers of Hamilton cycles in pseudorandom graphs, using the following comparatively weak pseudorandomness notion. A graph G is (ε, p, k, ℓ)-pseudorandom if for all disjoint X and Y ⊆ V (G) with |X| ≥ εp k n and |Y | ≥ εp ℓ n we have e(X, Y ) = (1 ± ε)p|X||Y |. We prove that for all β > 0 there is an ε > 0 such that an (ε, p, 1, 2)-pseudorandom graph on n vertices with minimum degree at least βpn contains the square of a Hamilton cycle. In particular, this implies that (n, d, λ)-graphs with λ ≪ d 5/2 n −3/2 contain the square of a Hamilton cycle, and thus a triangle factor if n is a multiple of 3. This improves on a result of Krivelevich, Sudakov and Szabó [Triangle factors in sparse pseudo-random graphs, Combinatorica 24 (2004), no. 3, 403-426].We also extend our result to higher powers of Hamilton cycles and establish corresponding counting versions.

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Section: Our Resultsmentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

Section: Enumerating Powers Of Hamilton Cyclesmentioning

confidence: 99%

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Powers of Hamilton cycles in pseudorandom graphs

et al. 2016

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“…When, later in the proof, we need to use some vertices from the reservoir set, we use the deterministic graph G α to swap out some vertices from the reservoir. Similar kind of reservoir structures were used for embedding bounded degree trees [18] and tight Hamilton cycles in hypergraphs [2], but we use the interplay of the random and deterministic graphs in a new way to create this structure.…”

Section: Preparing the Reservoirmentioning

confidence: 99%

Embedding spanning bounded degree subgraphs in randomly perturbed graphs

Böttcher

Montgomery

Parczyk

et al. 2017

Electronic Notes in Discrete Mathematics

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Original citation:Böttcher, Julia, Montgomery, Richard, Parczyk, Olaf, Person, Yury.(2017) This document is the author's final accepted version of the journal article. There may be differences between this version and the published version. You are advised to consult the publisher's version if you wish to cite from it.Embedding spanning bounded degree subgraphs in randomly perturbed graphs AbstractWe study the model of randomly perturbed dense graphs, which is the union of any graph G α with minimum degree αn and the binomial random graph G(n, p). For p = ω(n −2/(∆+1) ), we show that G α ∪ G(n, p) contains any single spanning graph with maximum degree ∆. As in previous results concerning this model, the bound for p we use is lower by a log-term in comparison to the bound known to be needed to find the same subgraph in G(n, p) alone.

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“…Using the second‐moment method, Dudek and Frieze have shown that

1 / n

is a threshold for the appearance of a tight Hamilton cycle in a random k ‐graph for any

k \geq 3

. An algorithmic proof for

p \geq n^{- 1 + ɛ}

was given by Allen, Böttcher, Kohayakawa and Person , for any constant

ɛ > 0

. The next theorem, again, lies between these two results: the bound on p is a logarithmic factor away from the optimal one and we obtain a quasi‐polynomial algorithm (instead of a polynomial one which requires p to be a factor of

n^{ɛ}

away from the smallest possible value, as in ).Theorem Let

k \geq 2

be an integer.…”

Section: Introductionmentioning

confidence: 99%

Powers of Hamilton cycles in random graphs and tight Hamilton cycles in random hypergraphs

Nenadov

Škorić

2018

Random Struct Algorithms

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We show that for every k∈N there exists C > 0 such that if pk≥Clog8n/n then asymptotically almost surely the random graph G(n,p) contains the kth power of a Hamilton cycle. This determines the threshold for appearance of the square of a Hamilton cycle up to the logarithmic factor, improving a result of Kühn and Osthus. Moreover, our proof provides a randomized quasi‐polynomial algorithm for finding such powers of cycles. Using similar ideas, we also give a randomized quasi‐polynomial algorithm for finding a tight Hamilton cycle in the random k‐uniform hypergraph G(k)(n,p) for p≥Clog8n/n. The proofs are based on the absorbing method and follow the strategy of Kühn and Osthus, and Allen et al. The new ingredient is a general Connecting Lemma which allows us to connect tuples of vertices using arbitrary structures at a nearly optimal value of p. Both the Connecting Lemma and its proof, which is based on Janson's inequality and a greedy embedding strategy, might be of independent interest.

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Tight Hamilton cycles in random hypergraphs

Cited by 20 publications

References 24 publications

Powers of Hamilton cycles in pseudorandom graphs

Powers of Hamilton cycles in pseudorandom graphs

Embedding spanning bounded degree subgraphs in randomly perturbed graphs

Powers of Hamilton cycles in random graphs and tight Hamilton cycles in random hypergraphs

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