Minimum codegree threshold for Hamilton ℓ-cycles in k-uniform hypergraphs

Han, Jie; Zhao, Yi

doi:10.1016/j.jcta.2015.01.004

Cited by 39 publications

(35 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Very recently Han and the author [33] determined the d-degree threshold h When k − l does not divide k, the threshold h l (k, n) is much smaller. Kühn and Osthus [51] proved that h 1 (3, n) = n/4 + o(n).…”

Section: Theorem 32 [74]mentioning

confidence: 99%

Recent advances on Dirac-type problems for hypergraphs

Zhao

2016

The IMA Volumes in Mathematics and Its Applications

Self Cite

View full text Add to dashboard Cite

A fundamental question in graph theory is to establish conditions that ensure a graph contains certain spanning subgraphs. Two well-known examples are Tutte's theorem on perfect matchings and Dirac's theorem on Hamilton cycles. Generalizations of Dirac's theorem, and related matching and packing problems for hypergraphs, have received much attention in recent years. New tools such as the absorbing method and regularity method have helped produce many new results, and yet some fundamental problems in the area remain unsolved. We survey recent developments on Dirac-type problems along with the methods involved, and highlight some open problems.Given two (hyper)graphs F and H, which conditions guarantee H contains F as a subgraph? When |V (F )| = |V (H)|, the decision problem of whether H contains F is often NP-complete, e.g., deciding if a graph H contains a Hamilton cycle is a well-known NPcomplete problem. Therefore it is natural to look for sufficient conditions for such problems. A classical result of Dirac [13] states that every graph on n ≥ 3 vertices with minimum degree n/2 contains a Hamilton cycle. Problems that relate the minimum degree (in general, minimum d-degree in hypergraphs) to the structure of the (hyper)graphs are often referred to as Dirac-type problems. The Dirac-type problems for hypergraphs have received much attention in recent years. In this survey we concentrate on three such problems: matching problems (Section 1), packing problems (Section 2), and Hamilton cycles (Section 3). Many problems in this survey were already considered in the survey of Rödl and Ruciński [67]. However, since this is a fast-growing area, there are new developments in the last few years and we will emphasize these new advances. Since we only consider Dirac-type problems, we do not discuss matching, packing, or Hamilton cycles in random or quasi-random hypergraphs. We also omit corresponding results in graphs and digraphs. Many results that we omit can be found in other surveys, e.g., Kühn and Osthus [53,55,56], and Gould [23, 24]. Matching problemsGiven k ≥ 2, a k-uniform hypergraph (k-graph) consists of a vertex set V and an edge set E, where each edge is a k-element subset (k-subset) of V . Thus a 2-graph is simply a graph. In this survey a hypergraph refers to a k-graph with k ≥ 3. Given a k-graph H with k ≥ 2, a matching of size s is a collection of s disjoint edges; a perfect matching is a matching that covers the vertex set of H (thus it is necessary that k divides |V (H)|). Many open problems in combinatorics can be formulated as a problem of finding perfect matchings in hypergraphs, e.g., Ryser's conjecture that every Latin square of odd order has a transversal, and the existence of combinatorial designs (recently solved by Keevash [41]).1991 Mathematics Subject Classification. 05C65, 05C70, 05C45, 05C35.

show abstract

Section: Theorem 32 [74]mentioning

confidence: 99%

Recent advances on Dirac-type problems for hypergraphs

Zhao

2016

The IMA Volumes in Mathematics and Its Applications

Self Cite

View full text Add to dashboard Cite

show abstract

“…Theorem 1.1 ([15, 24, 25, 26, 33, 34]). For any k ≥ 3, 1 ≤ < k and η > 0, there exists n 0 such that if n ≥ n 0 is divisible by k − and H is a k-graph on n vertices with More recently the exact Dirac threshold has been identified in some cases, namely for k = 3, = 2 by Rödl, Ruciński and Szemerédi [36], for k = 3, = 1 by Czygrinow and Molla [5], and for any k ≥ 3 and < k/2 by Han and Zhao [17].…”

Section: Introductionmentioning

confidence: 99%

Hamilton cycles in hypergraphs below the Dirac threshold

Garbe

Mycroft

2018

Journal of Combinatorial Theory, Series B

View full text Add to dashboard Cite

We establish a precise characterisation of 4-uniform hypergraphs with minimum codegree close to n/2 which contain a Hamilton 2-cycle. As an immediate corollary we identify the exact Dirac threshold for Hamilton 2-cycles in 4-uniform hypergraphs. Moreover, by derandomising the proof of our characterisation we provide a polynomial-time algorithm which, given a 4-uniform hypergraph H with minimum codegree close to n/2, either finds a Hamilton 2-cycle in H or provides a certificate that no such cycle exists. This surprising result stands in contrast to the graph setting, in which below the Dirac threshold it is NP-hard to determine if a graph is Hamiltonian. We also consider tight Hamilton cycles in k-uniform hypergraphs H for k ≥ 3, giving a series of reductions to show that it is NP-hard to determine whether a k-uniform hypergraph H with minimum degree δ(H) ≥ 1 2 |V (H)| − O(1) contains a tight Hamilton cycle. It is therefore unlikely that a similar characterisation can be obtained for tight Hamilton cycles.

show abstract

“…These results can all be collectively described by the following theorem, whose statement gives the asymptotic codegree Dirac threshold for any k and ℓ. Theorem 1.1 ([9, 14, 15, 16, 21, 22]). For any k ≥ 3, 1 ≤ ℓ < k and η > 0, there exists n 0 such that if n ≥ n 0 is divisible by k − ℓ and H is a k-graph on n vertices with More recently the exact codegree Dirac threshold (for large n) has been identified in some cases, namely for k = 3, ℓ = 2 by Rödl, Ruciński and Szemerédi [23], for k = 3, ℓ = 1 by Czygrinow and Molla [4], for any k ≥ 3 and ℓ < k/2 by Han and Zhao [10], and for k = 4 and ℓ = 2 by Garbe and Mycroft [7].…”

Section: Introductionmentioning

confidence: 99%

The minimum vertex degree for an almost-spanning tight cycle in a 3-uniform hypergraph

Cooley

Mycroft

2017

Discrete Mathematics

View full text Add to dashboard Cite

Abstract. We prove that any 3-uniform hypergraph whose minimum vertex degree is at least 5 9 + o(1) n 2 admits an almost-spanning tight cycle, that is, a tight cycle leaving o(n) vertices uncovered. The bound on the vertex degree is asymptotically best possible. Our proof uses the hypergraph regularity method, and in particular a recent version of the hypergraph regularity lemma proved by Allen, Böttcher, Cooley and Mycroft.

show abstract

Minimum codegree threshold for Hamilton ℓ-cycles in k-uniform hypergraphs

Abstract: For 1 ≤ < k/2, we show that for sufficiently large n, every k-uniform hypergraph on n vertices with minimum codegree at least n 2(k− ) contains a Hamilton -cycle. This codegree condition is best possible and improves on work of Hàn and Schacht who proved an asymptotic result.

Cited by 39 publications

References 22 publications

Recent advances on Dirac-type problems for hypergraphs

Recent advances on Dirac-type problems for hypergraphs

Hamilton cycles in hypergraphs below the Dirac threshold

The minimum vertex degree for an almost-spanning tight cycle in a 3-uniform hypergraph

Contact Info

Product

Resources

About