Minimum degree conditions for tight Hamilton cycles

Lang, Richard A.; Sanhueza‐Matamala, Nicolás

doi:10.48550/arxiv.2005.05291

Cited by 4 publications

(4 citation statements)

References 30 publications

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“…In previous articles written in collaboration with Ruciński, Schacht, and Szemerédi [15,17] we solved the cases k " 3 and k " 4. The general case was also obtained by Lang and Sanhueza-Matamala [13] in independent research. A construction due to Han and Zhao [11] reproduced in the introduction of [15] shows that the number 5 9 appearing in Theorem 1.2 is optimal.…”

Section: §1 Introductionmentioning

confidence: 73%

On Hamiltonian cycles in hypergraphs with dense link graphs

Polcyn,

Reiher,

Rödl

et al. 2020

Preprint

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Section: §1 Introductionmentioning

confidence: 73%

On Hamiltonian cycles in hypergraphs with dense link graphs

Polcyn,

Reiher,

Rödl

et al. 2020

Preprint

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“…It was recently proved independently by Lang and Sanhueza-Matamala [ 21 ] and by Polcyn, Reiher, Rödl, and Schülke [ 26 ] that every large k-graph on n vertices with δ k´2 pHq ě p5{9 òp1qq `n 2 ˘contains a Hamilton cycle. We wonder whether such k-graphs actually contain p1 óp1qq reg k pHq{k edge-disjoint Hamilton cycles.…”

Section: Denoting the Indicator Variable Of The Event D F Xcmentioning

confidence: 99%

Decomposing hypergraphs into cycle factors

Joos¹,

Kühn²,

Schülke³

2021

Preprint

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A famous result by Rödl, Ruciński, and Szemerédi guarantees a (tight) Hamilton cycle in k-uniform hypergraphs H on n vertices with minimum pk ´1q-degree δ k´1 pHq ě p1{2 `op1qqn, thereby extending Dirac's result from graphs to hypergraphs. For graphs, much more is known; each graph on n vertices with δpGq ě p1{2 `op1qqn contains p1 ´op1qqr edgedisjoint Hamilton cycles where r is the largest integer such that G contains a spanning 2r-regular subgraph, which is clearly asymptotically optimal. This was proved by Ferber, Krivelevich, and Sudakov answering a question raised by Kühn, Lapinskas, and Osthus.We extend this result to hypergraphs; every k-uniform hypergraph H on n vertices with δ k´1 pHq ě p1{2 `op1qqn contains p1 ´op1qqr edge-disjoint (tight) Hamilton cycles where r is the largest integer such that H contains a spanning subgraph with each vertex belonging to kr edges. In particular, this yields an asymptotic solution to a question of Glock, Kühn, and Osthus.In fact, our main result applies to approximately vertex-regular k-uniform hypergraphs with a weak quasirandom property and provides approximate decompositions into cycle factors without too short cycles.

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“…The proof was inspired by the proof of Lemma 8.8 in [11]. A different generalisation of this lemma can also be found as Lemma 2.3 in [13].…”

Section: Preliminariesmentioning

confidence: 99%

Towards Lehel's conjecture for 4-uniform tight cycles

Lo¹,

Pfenninger²

2020

Preprint

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A k-uniform tight cycle is a k-uniform hypergraph with a cyclic ordering of its vertices such that its edges are all the sets of size k formed by k consecutive vertices in the ordering. We prove that every red-blue edge-coloured K (4) n contains a red and a blue tight cycle that are vertex-disjoint and together cover n − o(n) vertices. Moreover, we prove that every red-blue edge-coloured K(5) n contains four monochromatic tight cycles that are vertex-disjoint and together cover n − o(n) vertices.

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Minimum degree conditions for tight Hamilton cycles

Cited by 4 publications

References 30 publications

On Hamiltonian cycles in hypergraphs with dense link graphs

On Hamiltonian cycles in hypergraphs with dense link graphs

Decomposing hypergraphs into cycle factors

Towards Lehel's conjecture for 4-uniform tight cycles

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