Perfect Matchings and K 4 3 -Tilings in Hypergraphs of Large Codegree

Pikhurko, Oleg

doi:10.1007/s00373-008-0787-7

Cited by 70 publications

(90 citation statements)

References 6 publications

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“…Lo and Markström [25,27] showed that δ(K 3 4 − e, n) = n/2 + o(n) and that δ(K 3 4 , n) = 3n/4 + o(n). Simultaneously with the latter, Keevash and Mycroft [16] showed that the exact value of δ(K 3 4 , n) for large n is 3n/4 − 1 or 3n/4 − 2, again according to the parity of n/4; these results confirmed a conjecture of Pikhurko [29], who had previously shown that δ(K 3 4 , n) ≤ 0.8603n, and who gave the construction which establishes the lower bound on δ(K 3 4 , n). The exact value of δ(K 3 4 − e, n) for large n remains an open problem.…”

Section: Perfect Packings In Graphssupporting

confidence: 62%

“…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%

“…The fact that δ(K 3 4 , n) ≤ 3n/4 strongly suggests that we have δ(K, n) ≤ 3n/4 + o(n) for any values of b 1 , b 2 , b 3 and b 4 . On the other hand, if gcd({b 1 , b 2 , b 3 , b 4 }) > 1, then we can modify a construction of Pikhurko [29] to show that δ(K, n) ≥ 3n/4−2, and in fact the same construction gives the same threshold if gcd(K) = 2, where gcd(K) := gcd({b i − b j : i, j ∈ [4]}) > 1. So we expect that δ(K, n) = 3n/4 + o(1) in these cases.…”

mentioning

confidence: 99%

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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: Perfect Packings In Graphssupporting

confidence: 62%

Section: Perfect Packings In Graphsmentioning

confidence: 99%

mentioning

confidence: 99%

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Packing k-partite k-uniform hypergraphs

Mycroft

2016

Journal of Combinatorial Theory, Series A

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“…We will use the following theorem of Pikhurko , stated here in a less general form. Theorem Let H be a 4‐partite 4‐graph with 4‐partition

V (H) = V_{1} \cup V_{2} \cup V_{3} \cup V_{4}

, where

| V_{1} | = \dots = | V_{4} | = m

.…”

Section: Proof Of Theorem 14mentioning

confidence: 99%

Tiling 3‐Uniform Hypergraphs With

Czygrinow

DeBiasio

Nagle

2013

Journal of Graph Theory

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“…The extremal constructions are similar to the parity based one of H above. This improves asymptotic bounds in [25,28,29]. For d < k/2 less is known.…”

Section: 2mentioning

confidence: 60%

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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Perfect Matchings and K 4 3 -Tilings in Hypergraphs of Large Codegree

Cited by 70 publications

References 6 publications

Packing k-partite k-uniform hypergraphs

Packing k-partite k-uniform hypergraphs

Tiling 3‐Uniform Hypergraphs With

Fractional and integer matchings in uniform hypergraphs

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