Factors and loose Hamilton cycles in sparse pseudo‐random hypergraphs

Hàn, Hiêp; Han, Jie; Morris, Patrick

doi:10.1002/rsa.21052

Cited by 4 publications

(4 citation statements)

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“…An extension of their results on F -factors for linear F to sparse quasi-random k-graphs was obtained recently by Hàn, Han and Morris [22].…”

Section: Introductionmentioning

confidence: 75%

$F$-factors in Quasi-random Hypergraphs

Ding¹,

Han²,

Sun³

et al. 2021

Preprint

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Given k ≥ 2 and two k-graphs (k-uniform hypergraphs) F and H, an F -factor in H is a set of vertex disjoint copies of F that together covers the vertex set of H. Lenz and Mubayi [J. Combin. Theory Ser. B, 2016 ] studied the F -factor problem in quasi-random k-graphs with minimum degree Ω(n k−1 ). They posed the problem of characterizing the k-graphs F such that every sufficiently large quasi-random k-graph with constant edge density and minimum degree Ω(n k−1 ) contains an F -factor, and in particular, they showed that all linear k-graphs satisfy this property.In this paper we prove a general theorem on F -factors which reduces the F -factor problem of Lenz and Mubayi to a natural sub-problem, that is, the F -cover problem. By using this result, we answer the question of Lenz and Mubayi for those F which are k-partite k-graphs, and for all 3-graphs F , separately. Our characterization result on 3-graphs is motivated by the recent work of Reiher, Rödl and Schacht [J. Lond. Math. Soc., 2018 ] that classifies the 3-graphs with vanishing Turán density in quasi-random k-graphs.

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“…An extension of their results on F -factors for linear F to sparse quasi-random k-graphs was obtained recently by Hàn, Han and Morris [22].…”

Section: Introductionmentioning

confidence: 75%

$F$-factors in Quasi-random Hypergraphs

Ding¹,

Han²,

Sun³

et al. 2021

Preprint

Self Cite

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show abstract

“…An extension of their results on

F

‐factors for linear

F

to sparse quasi‐random

k

‐ graphs was obtained recently by Hàn, Han, and Morris [23].…”

Section: Introductionmentioning

confidence: 92%

F$F$‐factors in Quasi‐random Hypergraphs

Ding

Han

Sun

et al. 2022

Journal of London Math Soc

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Given 𝑘 ⩾ 2 and two 𝑘-graphs (𝑘-uniform hypergraphs) 𝐹 and 𝐻, an 𝐹-factor in 𝐻 is a set of vertex-disjoint copies of 𝐹 that together cover the vertex set of 𝐻. Lenz and Mubayi [J. Combin. Theory Ser. B, 2016] studied the 𝐹-factor problem in quasi-random 𝑘-graphs with minimum degree Ω(𝑛 𝑘−1 ). They posed the problem of characterizing the 𝑘-graphs 𝐹 such that every sufficiently large quasi-random 𝑘-graph with constant edge density and minimum degree Ω(𝑛 𝑘−1 ) contains an 𝐹-factor, and, in particular, they showed that all linear 𝑘-graphs satisfy this property. In this paper we prove a general theorem on 𝐹-factors which reduces the 𝐹-factor problem of Lenz and Mubayi to a natural sub-problem, that is, the 𝐹cover problem. By using this result, we answer the question of Lenz and Mubayi for those 𝐹 which are 𝑘-partite 𝑘-graphs, and for all 3-graphs 𝐹, separately. Our characterization result on 3-graphs is motivated by the recent work of Reiher, Rödl, and Schacht [J. Lond. Math. Soc., 2018] that classifies the 3-graphs with vanishing Turán density in quasi-random 𝑘-graphs.

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“…Building on previous work [23,47,55,56] mainly concerned with dense hypergraphs (the so-called quasirandom regime), Hiê . p Hàn, Jie Han and the author [30,31] recently gave the best-known conditions on pseudorandomness that guarantee different linear subgraphs of hypergraphs. These include all fixed sized linear subgraphs as well as F -factors for linear F (including perfect matchings) and loose Hamilton cycles.…”

Section: A(i)mentioning

confidence: 99%

Clique factors in pseudorandom graphs

Morris¹

2021

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An n-vertex graph is said to to be (p, β)-bijumbled if for any vertex setsWe prove that for any 3 ≤ r ∈ N and c > 0 there exists an ε > 0 such that any n-vertex (p, β)-bijumbled graph with n ∈ rN, δ(G) ≥ cpn and β ≤ εp r−1 n, contains a K r -factor. This implies a corresponding result for the stronger pseudorandom notion of (n, d, λ)-graphs.For the case of triangle factors, that is when r = 3, this result resolves a conjecture of Krivelevich, Sudakov and Szabó from 2004 and it is tight due to a pseudorandom triangle-free construction of Alon. In fact, in this case even more is true: as a corollary to this result and a result of Han, Kohayakawa, Person and the author, we can conclude that the same condition of β = o(p 2 n) actually guarantees that a (p, β)-bijumbled graph G contains every graph on n vertices with maximum degree at most 2.

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Factors and loose Hamilton cycles in sparse pseudo‐random hypergraphs

Abstract: We investigate the emergence of subgraphs in sparse pseudo-random k-uniform hypergraphs, using the following comparatively weak notion of pseudo-randomness.

Cited by 4 publications

References 58 publications

$F$-factors in Quasi-random Hypergraphs

$F$-factors in Quasi-random Hypergraphs

F$F$‐factors in Quasi‐random Hypergraphs

Clique factors in pseudorandom graphs

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