Polynomial-time perfect matchings in dense hypergraphs

Keevash, Peter; Knox, Fiachra; Mycroft, Richard

doi:10.1016/j.aim.2014.10.009

Cited by 18 publications

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“…The following lemma states that if almost all (k − 1)-tuples of vertices of a k-graph H have high degree, then we can find a k-complex J which covers almost all of the vertices of H, such that J has a useful degree sequence and the 'top level' J = of J is a subgraph of H. The k-partite form of this lemma was given by Keevash, Knox and Mycroft [14,Lemma 7.3] with a straightforward proof. The proof of the form given below is identical except for the simplification of not having to handle multiple vertex classes, so we omit it (this form is also implicit in [16]).…”

Section: 6mentioning

confidence: 99%

“…In particular, in recent years there has been much study of the case of a perfect matching, see e.g. [1,2,6,11,14,16,17,18,23,26,28,29,35,36]. For perfect Hpackings other than a perfect matching, results are much more sparse.…”

Section: Perfect Packings In Graphsmentioning

confidence: 99%

“…One example of the latter category is the cycle C 3 3 , which has type 0 and for which Theorem 1.4 showed that δ(C 3 3 , n) = n/3 + o(n) = σ(C 3 3 )n + o(n). The principal difference between the proofs of Theorem 1.4 and Lemma 3.2 was the use of Proposition 8.2 to find copies of C 3 3 with an odd number of vertices on each side of a partition of V (H); it seems likely that the value of δ(K, n) for k-partite k-graphs K of type 0 or type d ≥ 2 which are not tightly k-partite depends on the minimum codegree required to give an analogue of Proposition 8.2, in a manner reminiscent of the results of [14] for perfect matchings.…”

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Packing k-partite k-uniform hypergraphs

Mycroft

2016

Journal of Combinatorial Theory, Series A

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Abstract. Let G and H be k-graphs (k-uniform hypergraphs); then a perfect H-packing in G is a collection of vertex-disjoint copies of H in G which together cover every vertex of G. For any fixed H let δ(H, n) be the minimum δ such that any k-graph G on n vertices with minimum codegree δ(G) ≥ δ contains a perfect H-packing. The problem of determining δ(H, n) has been widely studied for graphs (i.e. 2-graphs), but little is known for k ≥ 3. Here we determine the asymptotic value of δ(H, n) for all complete k-partite k-graphs H, as well as a wide class of other k-partite k-graphs. In particular, these results provide an asymptotic solution to a question of Rödl and Ruciński on the value of δ(H, n) when H is a loose cycle. We also determine asymptotically the codegree threshold needed to guarantee an H-packing covering all but a constant number of vertices of G for any complete k-partite k-graph H.

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Section: 6mentioning

confidence: 99%

Section: Perfect Packings In Graphsmentioning

confidence: 99%

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Packing k-partite k-uniform hypergraphs

Mycroft

2016

Journal of Combinatorial Theory, Series A

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“…Then G contains a perfect K r -packing. Keevash and Knox [14] also have announced a proof of Theorem 1.3 in the case when r = 3. Conjecture 1.2 considers a 'Pósa-type' degree sequence condition: Pósa's theorem [25] states that a graph G on n ≥ 3 vertices has a Hamilton cycle if its degree sequence d 1 ≤ • • • ≤ d n satisfies d i ≥ i+1 for all i < (n−1)/2 and if additionally d ⌈n/2⌉ ≥ ⌈n/2⌉ when n is odd.…”

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A degree sequence Hajnal–Szemerédi theorem

Treglown

2016

Journal of Combinatorial Theory, Series B

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We say that a graph G has a perfect H-packing if there exists a set of vertex-disjoint copies of H which cover all the vertices in G. The seminal Hajnal-Szemerédi theorem [12] characterises the minimum degree that ensures a graph G contains a perfect Kr-packing. Balogh, Kostochka and Treglown [4] proposed a degree sequence version of the Hajnal-Szemerédi theorem which, if true, gives a strengthening of the Hajnal-Szemerédi theorem. In this paper we prove this conjecture asymptotically. Another fundamental result in the area is the Alon-Yuster theorem [3] which gives a minimum degree condition that ensures a graph contains a perfect H-packing for an arbitrary graph H. We give a wide-reaching generalisation of this result by answering another conjecture of Balogh, Kostochka and Treglown [4] on the degree sequence of a graph that forces a perfect H-packing. We also prove a degree sequence result concerning perfect transitive tournament packings in directed graphs. The proofs blend together the regularity and absorbing methods.Date: September 22, 2018.1 Conjecture 1.2 (Balogh, Kostochka and Treglown [4]). Let n, r ∈ N such that r divides n. Suppose that G is a graph on n vertices with degree sequence d 1 ≤ · · · ≤ d n such that:(α) d i ≥ (r − 2)n/r + i for all i < n/r; (β) d n/r+1 ≥ (r − 1)n/r. Then G contains a perfect K r -packing.Note that Conjecture 1.2, if true, is much stronger than the Hajnal-Szemerédi theorem since the degree condition allows for n/r vertices to have degree less than (r − 1)n/r. Moreover, the degree sequence condition in Conjecture 1.2 is best-possible. Indeed, examples in Section 4 in [4] show that one cannot replace (α) with d i ≥ (r − 2)n/r + i − 1 for a single value of i < n/r and (β) cannot be replaced with d n/r+1 ≥ (r − 1)n/r − 1. Chvátal's theorem on Hamilton cycles [6] implies Conjecture 1.2 in the case when r = 2. In [4] it was shown that Conjecture 1.2 is true under the additional assumption that no vertex x ∈ V (G) of degree less than (r − 1)n/r lies in a copy of K r+1 .In this paper we prove the following asymptotic version of Conjecture 1.2.Theorem 1.3. Let γ > 0 and r ∈ N. Then there exists an integer n 0 = n 0 (γ, r) such that the following holds. Suppose that G is a graph on n ≥ n 0 vertices where r divides n and with degree sequence d 1 ≤ · · · ≤ d n such that:Keevash and Knox [14] also have announced a proof of Theorem 1.3 in the case when r = 3. Conjecture 1.2 considers a 'Pósa-type' degree sequence condition: Pósa's theorem [25] states that a graph G on n ≥ 3 vertices has a Hamilton cycle if its degree sequence d 1 ≤ · · · ≤ d n satisfies d i ≥ i+1 for all i < (n−1)/2 and if additionally d ⌈n/2⌉ ≥ ⌈n/2⌉ when n is odd. The aforementioned result of Chvátal [6] generalises Pósa's theorem by characterising those degree sequences which force Hamiltonicity. It would be interesting to establish an analogous result for perfect K r -packings.After this paper was submitted, the author and Staden [27] have used Theorem 1.3 to prove a degree sequence version of Pósa's conjecture on ...

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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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Polynomial-time perfect matchings in dense hypergraphs

Cited by 18 publications

References 30 publications

Packing k-partite k-uniform hypergraphs

Packing k-partite k-uniform hypergraphs

A degree sequence Hajnal–Szemerédi theorem

Recent advances on Dirac-type problems for hypergraphs

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