Exact minimum degree thresholds for perfect matchings in uniform hypergraphs

Treglown, Andrew; Zhao, Yi

doi:10.1016/j.jcta.2012.04.006

Cited by 70 publications

(78 citation statements)

References 28 publications

(55 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In many problems on finding H-factors such a property indeed holds (e.g. see [13,20,24]). However, as pointed out in [3], the conditions on G in the case r = 3 and ℓ = 2 in Theorem 1.2 are not strong enough to guarantee this property.…”

Section: Proof Strategymentioning

confidence: 99%

On a Ramsey--Turán Variant of the Hajnal--Szemerédi Theorem

Nenadov¹,

Pehova²

2020

SIAM J. Discrete Math.

View full text Add to dashboard Cite

A seminal result of Hajnal and Szemerédi states that if a graph G with n vertices has minimum degree δ(G) ≥ (r − 1)n/r for some integer r ≥ 2, then G contains a K r -factor, assuming r divides n. Extremal examples which show optimality of the bound on δ(G) are very structured and, in particular, contain large independent sets. In analogy to the Ramsey-Túran theory, Balogh, Molla, and Sharifzadeh initiated the study of how the absence of such large independent sets influences sufficient minimum degree. We show the following two related results:• For any r > ℓ ≥ 2, if G is a graph satisfying δ(G) ≥ r−ℓ r−ℓ+1 n+Ω(n) and α ℓ (G) = o(n), that is, a largest K ℓ -free induced subgraph has at most o(n) vertices, then G contains a K r -factor. This is optimal for ℓ = r − 1 and extends a result of Balogh, Molla, and Sharifzadeh who considered the case r = 3.• If a graph G satisfies δ(G) = Ω(n) and α * r (G) = o(n), that is, every induced K r -free r-partite subgraph of G has at least one vertex class of size o(n), then it contains a K r -factor. A similar statement is proven for a general graph

show abstract

Section: Proof Strategymentioning

confidence: 99%

On a Ramsey--Turán Variant of the Hajnal--Szemerédi Theorem

Nenadov¹,

Pehova²

2020

SIAM J. Discrete Math.

View full text Add to dashboard Cite

show abstract

“…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

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…This conjecture was proved recently by Han [3]. Treglown and Zhao [13,14] determined the minimum l-degree threshold for perfect matchings in k-graphs for k l k 2 − 1 ∕ ≤ ≤ . Bollobás, Daykin, and Erdős [2] considered the minimum vertex degree for the appearance of m-matchings.…”

mentioning

confidence: 83%