Matchings in 3-uniform hypergraphs

Kühn, Daniela; Osthus, Deryk; Treglown, Andrew

doi:10.1016/j.jctb.2012.11.005

Cited by 85 publications

(101 citation statements)

References 14 publications

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“…Khan [16] also determined m n/k 1 (k, n) exactly for k = 4. As a consequence of these results, m s 1 (k, n) is determined exactly whenever s ≤ n/k and k ≤ 4 (for details see the concluding remarks in [19]). More generally, we propose the following version of Conjecture 1.3 for non-perfect matchings.…”

Section: 2mentioning

confidence: 70%

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Fractional and integer matchings in uniform hypergraphs

Kühn

Osthus

Townsend

2014

European Journal of Combinatorics

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Abstract. A conjecture of Erdős from 1965 suggests the minimum number of edges in a kuniform hypergraph on n vertices which forces a matching of size t, where t ≤ n/k. Our main result verifies this conjecture asymptotically, for all t < 0.48n/k. This gives an approximate answer to a question of Huang, Loh and Sudakov, who proved the conjecture for t ≤ n/3k2 . As a consequence of our result, we extend bounds of Bollobás, Daykin and Erdős by asymptotically determining the minimum vertex degree which forces a matching of size t < 0.48n/(k − 1) in a k-uniform hypergraph on n vertices. We also obtain further results on d-degrees which force large matchings. In addition we improve bounds of Markström and Ruciński on the minimum ddegree which forces a perfect matching in a k-uniform hypergraph on n vertices. Our approach is to inductively prove fractional versions of the above results and then translate these into integer versions.

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Section: 2mentioning

confidence: 70%

“…In particular, this implies Conjecture 1.3 for k ≤ 5. Khan [15], and independently Kühn, Osthus and Treglown [19], determined m n/k 1 (k, n) exactly for k = 3. Khan [16] also determined m n/k 1 (k, n) exactly for k = 4.…”

Section: 2mentioning

confidence: 95%

Fractional and integer matchings in uniform hypergraphs

Kühn

Osthus

Townsend

2014

European Journal of Combinatorics

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“…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%

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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“…For 3‐graphs, Kühn et al. proved the following result. Theorem There exists a positive integer n 0 such that if H is a 3‐graph of order

n \geq n_{0}

, m is an integer with

1 \leq m \leq n / 3

, and

δ_{1} false(H false) > (\frac{0pt}{n - 1}) - (\frac{0pt}{n - m})

, then

ν (H) \geq m

.…”

Section: Introductionmentioning

confidence: 99%

Disjoint perfect matchings in 3‐uniform hypergraphs

Zhang

2017

Journal of Graph Theory

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For a hypergraph H, let δ1false(Hfalse) denote the minimum vertex degree in H. Kühn, Osthus, and Treglown proved that, for any sufficiently large integer n with n≡0(mod3), if H is a 3‐uniform hypergraph with order n and δ1false(Hfalse)>0ptn−12−0pt2n/32 then H has a perfect matching, and this bound on δ1false(Hfalse) is best possible. In this article, we show that under the same conditions, H contains at least ⌈(2n+3)/9⌉ pairwise disjoint perfect matchings, and this bound is sharp.

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Matchings in 3-uniform hypergraphs

Cited by 85 publications

References 14 publications

Fractional and integer matchings in uniform hypergraphs

Fractional and integer matchings in uniform hypergraphs

Packing k-partite k-uniform hypergraphs

Disjoint perfect matchings in 3‐uniform hypergraphs

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