The Size of a Hypergraph and its Matching Number

Huang, Hao; Loh, Po‐Shen; Sudakov, Benny

doi:10.1017/s096354831100068x

Cited by 120 publications

(110 citation statements)

References 9 publications

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“…Bollobás, Daykin and Erdős [4] proved Conjecture 1.1 for general k whenever s < n/(2k 3 ), which extended earlier results of Erdős [8]. Huang, Loh and Sudakov [14] proved it for s < n/(3k 2 ). The main result in this paper verifies Conjecture 1.1 asymptotically for matchings of any size up to almost half the size of a perfect matching.…”

Section: Large Matchings In Hypergraphs With Many Edgessupporting

confidence: 72%

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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: Large Matchings In Hypergraphs With Many Edgessupporting

confidence: 72%

“…The main result in this paper verifies Conjecture 1.1 asymptotically for matchings of any size up to almost half the size of a perfect matching. This gives an asymptotic answer to a question in [14]. Theorem 1.2.…”

Section: Large Matchings In Hypergraphs With Many Edgesmentioning

confidence: 83%

Fractional and integer matchings in uniform hypergraphs

Kühn

Osthus

Townsend

2014

European Journal of Combinatorics

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“…For k = 2 (graphs) the value of n 0 (2, t) was determined by Erdős and Gallai [2]. The case k = 3 was recently investigated by Frankl, Rödl, and Rucinśki [10] and n 0 (3, t) was finally determined by Luczak and Mieczkowska [21] for large t, and by Frankl [6] for all t. In general, Huang, Loh, and Sudakov [16] showed n 0 (k, t) < 3tk 2 , which was slightly improved in [9] and greatly improved to n 0 (k, t) ≤ (2t + 1)k − t by Frankl [7]. Frankl [5] showed that for every n, k, t if a k-graph F on [n] has no t + 1 pairwise disjoint edges then |F| ≤ t n−1 k−1 .…”

Section: Given Setsmentioning

confidence: 99%

Hypergraph Turán numbers of linear cycles

Füredi

Jiang

2014

Journal of Combinatorial Theory, Series A

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A k-uniform linear cycle of length ℓ, denoted by C (k) ℓ , is a cyclic list of k-sets A 1 , . . . , A ℓ such that consecutive sets intersect in exactly one element and nonconsecutive sets are disjoint. For all k ≥ 5 and ℓ ≥ 3 and sufficiently large n we detemine the largest size of a k-uniform set family on [n] not containing a linear cycle of length ℓ. For odd ℓ = 2t + 1 the unique extremal family F S consists of all k-sets in [n] intersecting a fixed t-set S in [n]. For even ℓ = 2t + 2, the unique extremal family consists of F S plus all the k-sets outside S containing some fixed two elements. For k ≥ 4 and large n we also establish an exact result for so-called minimal cycles. For all k ≥ 4 our results substantially extend Erdős' result on largest k-uniform families without t + 1 pairwise disjoint members and confirm, in a stronger form, a conjecture of Mubayi and Verstraëte [23]. Our main method is the delta system method.

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“…Hypergraphs model more general types of relations than graphs. For more results on hypergraph, we refer to [1,2,5,[11][12][13]. The matching polynomial of a graph was introduced by Farrell in 1979.…”

Section: Introductionmentioning

confidence: 99%

On the matching polynomial of hypergraphs

Guo¹,

Zhao²,

Mao³

2017

JACODESMATH

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The concept of the matching polynomial of a graph, introduced by Farrell in 1979, has received considerable attention and research. In this paper, we generalize this concept and introduce the matching polynomial of hypergraphs. A recurrence relation of the matching polynomial of a hypergraph is obtained. The exact matching polynomials of some special hypergraphs are given. Further, we discuss the zeros of matching polynomials of hypergraphs.2010 MSC: 05C31, 05C65, 05C70

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The Size of a Hypergraph and its Matching Number

Cited by 120 publications

References 9 publications

Fractional and integer matchings in uniform hypergraphs

Fractional and integer matchings in uniform hypergraphs

Hypergraph Turán numbers of linear cycles

On the matching polynomial of hypergraphs

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