Tight Co-Degree Condition for Perfect Matchings in 4-Graphs

Czygrinow, Andrzej; Kamat, Vikram

doi:10.37236/2338

Cited by 36 publications

(53 citation statements)

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“…Now assume that r = 3 and consider any U ⊆ V (H) \ V 0 with i P (U ) = (a 1 , a 2 , a 3 ) for some nonnegative integers a 1 + a 2 + a 3 = k + 1. Pick any (b 1 , b 2 , b 3 ) ∈ I µ P (H) and let c j = a j − b j for j ∈ [3]. Note that exactly one of the three consecutive integers Given a set S of k + 2 vertices, we call an edge e ∈ E(H) disjoint from S S-absorbing if there are two disjoint edges e 1 and e 2 in E(H) such that |e 1 ∩ S| = k − 2, |e 1 ∩ e| = 2, |e 2 ∩ S| = 4, and |e 2 ∩ e| = k − 4.…”

Section: 4mentioning

confidence: 99%

Near Perfect Matchings in ${k}$-Uniform Hypergraphs II

Han¹

2016

SIAM J. Discrete Math.

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Abstract. Suppose k ∤ n and H is an n-vertex k-uniform hypergraph. A near perfect matching in H is a matching of size ⌊n/k⌋. We give a divisibility barrier construction that prevents the existence of near perfect matchings in H. This generalizes the divisibility barrier for perfect matchings. We give a conjecture on the minimum d-degree threshold forcing a (near) perfect matching in H which generalizes a well-known conjecture on perfect matchings. We also verify our conjecture for various cases. Our proof makes use of the lattice-based absorbing method that the author used recently to solve two other problems on matching and tilings for hypergraphs.

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Section: 4mentioning

confidence: 99%

Near Perfect Matchings in ${k}$-Uniform Hypergraphs II

Han¹

2016

SIAM J. Discrete Math.

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“…Let H 2 be the k-graph whose edges are all k-sets that intersect S. Then H 2 does not contain a perfect matching and δ ℓ (H 2 ) = 1 − 1 − 1 k k−ℓ + o(1) n k−ℓ for all 1 ≤ ℓ ≤ k − 1. In recent years Conjecture 1.1 (and its exact counterpart) has received substantial attention [1,4,9,11,21,22,27,31,34,36,37,39,40,43,44,45]. In particular, the exact threshold is known for all ℓ such that 0.42k ≤ ℓ ≤ k − 1 as well as for a handful of other values of (k, ℓ).…”

Section: Introductionmentioning

confidence: 99%

The complexity of perfect matchings and packings in dense hypergraphs

Han

Treglown

2020

Journal of Combinatorial Theory, Series B

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Given two k-graphs H and F , a perfect F -packing in H is a collection of vertexdisjoint copies of F in H which together cover all the vertices in H. In the case when F is a single edge, a perfect F -packing is simply a perfect matching. For a given fixed F , it is often the case that the decision problem whether an n-vertex k-graph H contains a perfect F -packing is NP-complete. Indeed, if k ≥ 3, the corresponding problem for perfect matchings is NP-complete [17,7] whilst if k = 2 the problem is NP-complete in the case when F has a component consisting of at least 3 vertices [14].In this paper we give a general tool which can be used to determine classes of (hyper)graphs for which the corresponding decision problem for perfect F -packings is polynomial time solvable. We then give three applications of this tool: (i) Given 1 ≤ ℓ ≤ k − 1, we give a minimum ℓ-degree condition for which it is polynomial time solvable to determine whether a k-graph satisfying this condition has a perfect matching; (ii) Given any graph F we give a minimum degree condition for which it is polynomial time solvable to determine whether a graph satisfying this condition has a perfect F -packing; (iii) We also prove a similar result for perfect K-packings in k-graphs where K is a k-partite k-graph.For a range of values of ℓ, k (i) resolves a conjecture of Keevash, Knox and Mycroft [20] whilst (ii) answers a question of Yuster [47] in the negative. In many cases our results are best possible in the sense that lowering the minimum degree condition means that the corresponding decision problem becomes NP-complete.

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“…Using the absorbing method, Rödl, Ruciński and Szemerédi [70,72] showed that (9) h k−1 (k, n) = n 2 + o(n) for all k ≥ 3. With long and involved arguments, they [74] were able to obtain an exact result when k = 3.…”

Section: Hamilton Cyclesmentioning

confidence: 99%

“…Assume that 1 ≤ l < k and k − l divides n. Since every (k − 1)-cycle of order n contains an l-cycle on the same vertices, (9) implies that h l (k, n) ≤ n 2 + o(n). On the other hand, it is not hard to see that h l (k, n) ≥ n 2 − k when k − l divides k. In fact, when k divides n, a Hamilton l-cycle contains a perfect matching thus Construction 1.1 provides this bound; when k does not divide n, this was proven by Markström and Ruciński [64, Proposition 2].…”

Section: Theorem 32 [74]mentioning

confidence: 99%

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Recent advances on Dirac-type problems for hypergraphs

Zhao

2016

The IMA Volumes in Mathematics and Its Applications

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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.

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Tight Co-Degree Condition for Perfect Matchings in 4-Graphs

Abstract: We will give a tight minimum co-degree condition for a 4-uniform hypergraph to contain a perfect matching.

Cited by 36 publications

References 6 publications

Near Perfect Matchings in ${k}$-Uniform Hypergraphs II

Near Perfect Matchings in ${k}$-Uniform Hypergraphs II

The complexity of perfect matchings and packings in dense hypergraphs

Recent advances on Dirac-type problems for hypergraphs

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