# Towards a hypergraph version of the P\'osa-Seymour conjecture

@inproceedings{PavezSigne2021TowardsAH, title={Towards a hypergraph version of the P\'osa-Seymour conjecture}, author={Mat'ias Pavez-Sign'e and Nicol{\'a}s Sanhueza-Matamala and Maya Jakobine Stein}, year={2021} }

We prove that for fixed r > k > 2, every k-uniform hypergraph on n vertices having minimum codegree at least (1 − ( ( r−1 k−1 ) + ( r−2 k−2 ) )−1)n + o(n) contains the (r − k + 1)th power of a tight Hamilton cycle. This result may be seen as a step towards a hypergraph version of the Pósa–Seymour conjecture. Moreover, we prove that the same bound on the codegree suffices for finding a copy of every spanning hypergraph of tree-width less than r which admits a tree decomposition where every…

## References

SHOWING 1-10 OF 33 REFERENCES

On the Pósa-Seymour conjecture

- Computer ScienceJ. Graph Theory
- 1998

The following approximate version of Paul Seymour's conjecture is proved: for any > 0 and positive integer k, there is an n0 such that, if G has order n ≥ n0 and minimum degree at least ( k k+1 + )n, then G contains the kth power of a Hamilton cycle.

Squares of Hamiltonian cycles in 3-uniform hypergraphs

- Mathematics, Computer ScienceRandom Struct. Algorithms
- 2020

It is shown that every $3-uniform hypergraph with minimum pair degree at least n contains a squared Hamiltonian cycle, a first step towards a hypergraph version of the Posa-Seymour conjecture.

Proof of the bandwidth conjecture of Bollobás and Komlós

- Mathematics
- 2009

In this paper we prove the following conjecture by Bollobás and Komlós: For every γ > 0 and integers r ≥ 1 and Δ, there exists β > 0 with the following property. If G is a sufficiently large graph…

proof of a Packing Conjecture of Bollobás

- Mathematics, Computer ScienceComb. Probab. Comput.
- 1995

Here it is proved that for any positive integer Δ and real 0 c n 0, T is a tree of order n and maximum degree Δ, and G is a graph ofOrder n andmaximum degree not exceeding cn , then there is a packing of T and G.

Perfect Matchings and K43-Tilings in Hypergraphs of Large Codegree

- Mathematics, Computer ScienceGraphs Comb.
- 2008

AbstractFor a k-graph F, let tl(n, m, F) be the smallest integer t such that every k-graph G on n vertices in which every l-set of vertices is included in at least t edges contains a collection of…

Tight cycles and regular slices in dense hypergraphs

- Computer Science, MathematicsJ. Comb. Theory, Ser. A
- 2017

It is argued that random subcomplexes of partitions returned by (a suitable form of) the Strong Hypergraph Regularity Lemma capture many important structural properties of the original hypergraph, and are advocated for use in extremal hypergraph theory.

An approximate Dirac-type theorem for k-uniform hypergraphs

- Computer Science, MathematicsComb.
- 2008

An approximate version of an analogous result for uniform hypergraphs is proved: for every K ≥ 3 and γ > 0, and for all n large enough, a sufficient condition for an n-vertex k-uniform hypergraph to be hamiltonian is that each (k − 1)-element set of vertices is contained in at least (1/2 + γ)n edges.

On Perfect Matchings in k-Complexes

- Mathematics
- 2019

Keevash and Mycroft [\emph{Mem.~Amer.~Math.~Soc., 2015}] developed a geometric theory for hypergraph matchings and characterized the dense simplicial complexes that contain a perfect matching. Their…

Some results on tree decomposition of graphs

- Mathematics, Computer ScienceJ. Graph Theory
- 1995

It is proved that the dilation of a graph is bounded by a logarithmic function of the congestion of the graph thereby settling a generalization of a conjecture of Bienstock.

Bandwidth, expansion, treewidth, separators and universality for bounded-degree graphs

- Mathematics, Computer ScienceEur. J. Comb.
- 2010

Relations between the bandwidth and the treewidth of bounded degree graphs G are established and it is shown that for each @c>0 every n-vertex graph with minimum degree ([email protected])n contains a copy of every bounded-degree planar graph on n vertices if n is sufficiently large.