Squares of Hamiltonian cycles in 3‐uniform hypergraphs

Bedenknecht, Wiebke; Reiher, Christian

doi:10.1002/rsa.20876

Cited by 3 publications

(2 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…With a slightly different but essentially equivalent notion of inseparability this was proved in 16 and in 1, but for the sake of completeness we provide the argument.…”

Section: Uniformly Dense and Inseparable Graphsmentioning

confidence: 94%

Embedding spanning subgraphs in uniformly dense and inseparable graphs

Ebsen

Maesaka

Reiher

et al. 2020

Random Struct Algorithms

Self Cite

View full text Add to dashboard Cite

We consider sufficient conditions for the existence of kth powers of Hamiltonian cycles in n-vertex graphs G with minimum degree n for arbitrarily small > 0. About 20 years ago Komlós, Sarközy, and Szemerédi resolved the conjectures of Pósa and Seymour and obtained optimal minimum degree conditions for this problem by showing that = k k+1 suffices for large n. For smaller values of the given graph G must satisfy additional assumptions. We show that inducing subgraphs of density > 0 on linear subsets of vertices and being inseparable, in the sense that every cut has density at least > 0, are sufficient assumptions for this problem and, in fact, for a variant of the bandwidth theorem. This generalizes recent results of Staden and Treglown. KEYWORDS absorption method, bandwidth theorem, powers of Hamiltonian cycles 1 INTRODUCTION We study sufficient conditions for the existence of spanning subgraphs in large finite graphs and begin the discussion with powers of Hamiltonian cycles. For k ∈ N the kth power of a given graph H is the graph H k on the same vertex set with xy being an edge in H k if x and y are distinct vertices of H that are connected in H by a path of at most k edges. For simplicity, we refer to a kth power of a path with at least k vertices as a k-path. Moreover, we refer to the ordered k-tuples of the first and last k vertices This is an open access article under the terms of the Creative Commons Attribution-NonCommercial License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited and is not used for commercial purposes.

show abstract

“…With a slightly different but essentially equivalent notion of inseparability this was proved in 16 and in 1, but for the sake of completeness we provide the argument.…”

Section: Uniformly Dense and Inseparable Graphsmentioning

confidence: 94%

Embedding spanning subgraphs in uniformly dense and inseparable graphs

Ebsen

Maesaka

Reiher

et al. 2020

Random Struct Algorithms

Self Cite

View full text Add to dashboard Cite

show abstract

“…For the existence of the

r\mathrm{th}

power of a tight Hamilton cycle in

k

‐graphs, not much is known in general. Bedenknecht and Reiher [6] showed that any 3‐graph on

n

vertices with codegree at least

\left(4/5+o(1)\right)n

contains the square of a tight Hamilton cycle, which was extended in [42] to arbitrary

k\ge 2

and arbitrary

r\ge 2

.…”

Section: Introductionmentioning

confidence: 99%

On powers of tight Hamilton cycles in randomly perturbed hypergraphs

Chang

Han

Thoma

2023

Random Struct Algorithms

View full text Add to dashboard Cite

For integers and , we show that for every , there exists such that the union of ‐uniform hypergraph on vertices with minimum codegree at least and a binomial random ‐uniform hypergraph with on the same vertex set contains the power of a tight Hamilton cycle with high probability. Moreover, a construction shows that one cannot take , where is a constant. Thus the bound on is optimal up to the value of and this answers a question of Bedenknecht, Han, Kohayakawa, and Mota.

show abstract

Counting spanning subgraphs in dense hypergraphs

Montgomery,

Pavez-Signé

2024

Combinator. Probab. Comp.

View full text Add to dashboard Cite

We give a simple method to estimate the number of distinct copies of some classes of spanning subgraphs in hypergraphs with a high minimum degree. In particular, for each $k\geq 2$ and $1\leq \ell \leq k-1$ , we show that every $k$ -graph on $n$ vertices with minimum codegree at least \begin{equation*} \left \{\begin {array}{l@{\quad}l} \left (\dfrac {1}{2}+o(1)\right )n & \text { if }(k-\ell )\mid k,\\[5pt] \left (\dfrac {1}{\lceil \frac {k}{k-\ell }\rceil (k-\ell )}+o(1)\right )n & \text { if }(k-\ell )\nmid k, \end {array} \right . \end{equation*} contains $\exp\!(n\log n-\Theta (n))$ Hamilton $\ell$ -cycles as long as $(k-\ell )\mid n$ . When $(k-\ell )\mid k$ , this gives a simple proof of a result of Glock, Gould, Joos, Kühn, and Osthus, while when $(k-\ell )\nmid k$ , this gives a weaker count than that given by Ferber, Hardiman, and Mond, or when $\ell \lt k/2$ , by Ferber, Krivelevich, and Sudakov, but one that holds for an asymptotically optimal minimum codegree bound.

show abstract

Squares of Hamiltonian cycles in 3‐uniform hypergraphs

Cited by 3 publications

References 13 publications

Embedding spanning subgraphs in uniformly dense and inseparable graphs

Embedding spanning subgraphs in uniformly dense and inseparable graphs

On powers of tight Hamilton cycles in randomly perturbed hypergraphs

Counting spanning subgraphs in dense hypergraphs

Contact Info

Product

Resources

About