Hamilton cycles in quasirandom hypergraphs

Lenz, John; Mubayi, Dhruv; Mycroft, Richard

doi:10.1002/rsa.20638

Cited by 9 publications

(21 citation statements)

References 27 publications

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“…Returning to Hamiltonicity, one encounters the following remarkable result of [27] stated here for 3-graphs only. Theorem 1.1 ([27]).…”

Section: Introductionmentioning

confidence: 87%

“…An additional result relevant to our account is that of [27]. Presentation of the latter requires a brief overview regarding quasirandom 3-graphs.…”

Section: Introductionmentioning

confidence: 99%

“…c For tight cycles, however, a result analogous to Theorem 1.1 does not exist for -quasirandom 3-graphs. Indeed, [27,Proposition 4] asserts that for every ρ > 0 and a sufficiently large n, an n-vertex (ρ, 1/8) -quasirandom 3-graph H exists satisfying δ(H) (1/8 − ρ) n−1 2 and having no tight Hamilton cycle. The constant 1/8 here is not best possible, though, as the following construction demonstrates.…”

Section: Introductionmentioning

confidence: 99%

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Tight Hamilton cycles in cherry-quasirandom 3-uniform hypergraphs

Aigner-Horev

Levy

2020

Combinator. Probab. Comp.

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We employ the absorbing-path method in order to prove two results regarding the emergence of tight Hamilton cycles in the so-called two-path or cherry-quasirandom 3-graphs. Our first result asserts that for any fixed real α > 0, cherry-quasirandom 3-graphs of sufficiently large order n having minimum 2-degree at least α(n – 2) have a tight Hamilton cycle. Our second result concerns the minimum 1-degree sufficient for such 3-graphs to have a tight Hamilton cycle. Roughly speaking, we prove that for every d, α > 0 satisfying d + α > 1, any sufficiently large n-vertex such 3-graph H of density d and minimum 1-degree at least $\alpha \left({\matrix{{n - 1} \cr 2 \cr } } \right)$ has a tight Hamilton cycle.

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“…Returning to Hamiltonicity, one encounters the following remarkable result of [27] stated here for 3-graphs only. Theorem 1.1 ([27]).…”

Section: Introductionmentioning

confidence: 87%

“…An additional result relevant to our account is that of [27]. Presentation of the latter requires a brief overview regarding quasirandom 3-graphs.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Tight Hamilton cycles in cherry-quasirandom 3-uniform hypergraphs

Aigner-Horev

Levy

2020

Combinator. Probab. Comp.

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“…The path absorbers, however, are more involved and differ from those used in absorbing arguments before. In particular, the first absorbers [7,22] for finding loose Hamilton cycles in hypergraphs do not satisfy our definition of path absorbers, while the ones used, e.g., in [35], have edge degeneracy k and so are not as effective as those given here, which have degeneracy k − 1.…”

Section: Absorbers and The Templatementioning

confidence: 84%

“…The random graph G(n, p) is p, O( √ pn) -jumbled almost always, which is essentially opti-statement is known but some natural linear spanning subgraphs have been studied. Lenz and Mubayi [34] and Lenz, Mubayi, and Mycroft [35] investigated the existence of perfect matchings, Ffactors for linear F , and loose Hamilton cycles. As usual an F -factor in a k-graph H is a collection of vertex disjoint copies of F in H which cover all of V (H).…”

Section: Introductionmentioning

confidence: 99%

Factors and loose Hamilton cycles in sparse pseudo-random hypergraphs

Hàn¹,

Han²,

Morris³

2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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We investigate the emergence of spanning structures in sparse pseudo-random kuniform hypergraphs, using the following comparatively weak notion of pseudo-randomness. A k-uniform hypergraph H on n vertices is called (p, α, ε)-pseudo-random if for all (not necessarily disjoint) vertex subsets A1, . . . , A k ⊆V (H) with |A1| · · · |A k |≥αn k we have e(A1, . . . , A k ) = (1 ± ε)p|A1| · · · |A k |.For any linear k-uniform F we provide a bound on α = α(n) in terms of p = p(n) and F , such that (under natural divisibility assumptions on n) any k-uniform p, α, o(1) -pseudo-random n-vertex hypergraph H with a mild minimum vertex degree condition contains an F -factor. The approach also enables us to establish the existence of loose Hamilton cycles in sufficiently pseudo-random hypergraphs and all results imply corresponding bounds for stronger notions of hypergraph pseudorandomness such as jumbledness or large spectral gap.As a consequence, p, α, o(1) -pseudo-random k-graphs as above contain: (i) a perfect matching if α = o(p k ) and (ii) a loose Hamilton cycle if α = o(p k−1 ). This extends the works of Lenz-Mubayi, and Lenz-Mubayi-Mycroft who studied the analogous problems in the dense setting.mal since any graph with edge density, say, p < 0.99 satisfies β = Ω( √ pn) (this follows from the proof of [15], see also [29]). /1, project ID: 390685689). 1 Throughout the paper we write x = y ± z to denote that y − z ≤ x ≤ y + z.1 One topic of great importance and popularity in the area concerns the appearance of certain subgraphs F in sufficiently pseudo-random graphs G. Here, F can be a small, fixed size graph such as a triangle, an odd cycle or a fixed size clique, or it can be a large, indeed spanning subgraph of G such as a perfect matching, a Hamilton cycle, or a K r -factor 2 . The fundamental question then concerns the degree of pseudo-randomness which ensures that F is a subgraph of G and we distinguish here the (dense) quasi-random case, when p = Ω(1) and β = o(n), and the (sparse) pseudo-random case, when p = o(1) and β = β(n, p) is a function of n and p. For the quasi-random case the subgraph containment problem is well understood [11,28]; for the pseudo-random case, however, it turned out to be notoriously difficult already for small graphs F , and even more so for spanning subgraphs. Thus, while bounds exist for many classes of graphs F (see e.g. [3]), only few are known to be (essentially) best possible: triangles, odd cycles, perfect matchings, Hamilton cycles and triangle-factors [4,5,29,38].(Linear) pseudo-random hypergraphs. In this paper we investigate the corresponding question for hypergraphs. A k-uniform hypergraph, k-graph for short, is a pair H = (V, E) with a vertex set V = V (H) and an edge set E = E(H)⊆ V k , where V k denotes the set of all k-element subsets of V . Launched by Chung and Graham [12] the investigation of pseudo-random k-graphs is widely popular, albeit mostly restricted to the dense case [1, 8-10, 13, 20, 21, 26, 27, 32, 33, 40-42, 47]. There are several generalisations of (1.1) t...

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