Tiling 3‐Uniform Hypergraphs With

Czygrinow, Andrzej; DeBiasio, Louis; Nagle, Brendan

doi:10.1002/jgt.21726

Cited by 27 publications

(76 citation statements)

References 11 publications

(31 reference statements)

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“…[6, Lemma 3.3] For all γ, δ > 0, there exists α > 0 so that the following holds for all sufficiently large n ∈ 4N. Let H be an n-vertex 3-graph with δ 2 (H) ≥ δn.…”

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confidence: 99%

On Perfect Matchings and Tilings in Uniform Hypergraphs

Han¹

2018

SIAM J. Discrete Math.

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In this paper we study some variants of Dirac-type problems in hypergraphs. First, we show that for k ≥ 3, if H is a k-graph on n ∈ kN vertices with independence number at most n/p and minimum codegree at least (1/p+o(1))n, where p is the smallest prime factor of k, then H contains a perfect matching. Second, we show that if H is a 3-graph on n ∈ 3N vertices which does not contain any induced copy of K − 4 (the unique 3-graph with 4 vertices and 3 edges) and has minimum codegree at least (1/3 + o(1)))n, then H contains a perfect matching. Moreover, if we allow the matching to miss at most 3 vertices, then the minimum degree condition can be reduced to (1/6+o(1)))n. Third, we show that if H is a 3-graph on n ∈ 4N vertices which does not contain any induced copy of K − 4 and has minimum codegree at least (1/8 + o(1)))n, then H contains a perfect Y -tiling, where Y represents the unique 3-graph with 4 vertices and 2 edges. We also provide the examples showing that our minimum codegree conditions are asymptotically best possible. Our main tool for finding the perfect matching is a characterization theorem in [17] that characterizes the k-graphs with minimum codegree at least n/k which contain a perfect matching.

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“…[6, Lemma 3.3] For all γ, δ > 0, there exists α > 0 so that the following holds for all sufficiently large n ∈ 4N. Let H be an n-vertex 3-graph with δ 2 (H) ≥ δn.…”

mentioning

confidence: 99%

On Perfect Matchings and Tilings in Uniform Hypergraphs

Han¹

2018

SIAM J. Discrete Math.

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“…Let us introduce the 3-graphs relevant to the present work. Let K t = ([t], [t] (3) ) denote the complete 3-graph on t vertices, and let K − t denote the 3-graph obtained from K t by removing one 3-edge. The strong or tight t-cycle is the 3-graph C t on [t] with 3-edges {123, 234, 345, .…”

Section: The Problemmentioning

confidence: 99%

Codegree Thresholds for Covering 3-Uniform Hypergraphs

Falgas–Ravry¹,

Zhao²

2016

SIAM J. Discrete Math.

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“…where c ∈ {2, 3}; for k = 3 and ℓ = 2, Kühn and Osthus [6] showed that t 2 (n, Y 3,2 ) = n/4 + o(n), the exact value of t 2 (n, Y 3,2 ) was given by Czygrinow, DeBiasio and Nagle [1]; as a generalization, Gao, Han and Zhao [3] determined the exact value of t k−1 (n, Y k,ℓ ) for all k ≥ 3 and ℓ = k − 1, i.e. they proved that for any k ≥ 3 and sufficiently large n divisible by k + 1,…”

Section: Introductionmentioning

confidence: 99%

Codegree Threshold for Tiling k-graphs with Two Edges Sharing Exactly ℓ Vertices

Hou

2019

Acta. Math. Sin.-English Ser.

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Given integer k and a k-graph F , let t k−1 (n, F ) be the minimum integer t such that every k-graph H on n vertices with codegree at least t contains an F -factor. For integers k ≥ 3 and 0 ≤ ℓ ≤ k − 1, let Y k,ℓ be a k-graph with two edges that shares exactly ℓ vertices. Han and Zhao (JCTA, 2015) asked the following question: For all k ≥ 3, 0 ≤ ℓ ≤ k − 1 and sufficiently large n divisible by 2k − ℓ, determine the exact value of t k−1 (n, Y k,ℓ ). In this paper, we show that t k−1 (n, Y k,ℓ ) = n 2k−ℓ for k ≥ 3 and 1 ≤ ℓ ≤ k − 2, combining with two previously known results of Rödl, Ruciński and Szemerédi (JCTA, 2009) and Gao, Han and Zhao (arXiv, 2016), the question of Han and Zhao is solved completely.

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Tiling 3‐Uniform Hypergraphs With

Cited by 27 publications

References 11 publications

On Perfect Matchings and Tilings in Uniform Hypergraphs

On Perfect Matchings and Tilings in Uniform Hypergraphs

Codegree Thresholds for Covering 3-Uniform Hypergraphs

Codegree Threshold for Tiling k-graphs with Two Edges Sharing Exactly ℓ Vertices

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