Most Probably Intersecting Hypergraphs

Das, Shagnik; Sudakov, Benny

doi:10.37236/4784

Cited by 6 publications

(4 citation statements)

References 12 publications

(28 reference statements)

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“…Clearly, H is intersecting if and only if α (H) = 1. Intersecting hypergraphs are well studied in the literature (see, for example, [3,8,10,11,13,19,20,24]). For an intersecting hypergraph H of rank r, we immediately have γ(H) ≤ r − 1.…”

Section: Domination Matchings and Transversals In Hypergraphsmentioning

confidence: 99%

Domination in intersecting hypergraphs

Dong

Shan

Kang

et al. 2018

Discrete Applied Mathematics

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A matching in a hypergraph H is a set of pairwise disjoint hyperedges. The matching number α (H) of H is the size of a maximum matching in H. A subset D of vertices of H is a dominating set of H if for every v ∈ V \ D there exists u ∈ D such that u and v lie in an hyperedge of H. The cardinality of a minimum dominating set of H is called the domination number of H, denoted by γ(H). It is known that for a intersecting hypergraph H with rank r, γ(H) ≤ r − 1. In this paper we present structural properties on intersecting hypergraphs with rank r satisfying the equality γ(H) = r − 1. By applying the properties we show that all linear intersecting hypergraphs H with rank 4 satisfying γ(H) = r − 1 can be constructed by the well-known Fano plane.

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Section: Domination Matchings and Transversals In Hypergraphsmentioning

confidence: 99%

Domination in intersecting hypergraphs

Dong

Shan

Kang

et al. 2018

Discrete Applied Mathematics

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“…Das, Gan and Sudakov [7] studied the supersaturation problem, determining the minimum number of disjoint pairs appearing in sufficiently sparse k-uniform families. Furthermore, a probabilistic variant of this supersaturation problem was introduced by Katona, Katona and Katona [21], and further studied by Russell [25], Russell and Walters [26] and Das and Sudakov [8].…”

Section: Introductionmentioning

confidence: 99%

Removal and Stability for Erdös--Ko--Rado

Das

Tran

2016

SIAM J. Discrete Math.

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A k-uniform family of subsets of [n] is intersecting if it does not contain a disjoint pair of sets. The study of intersecting families is central to extremal set theory, dating back to the seminal Erdős-Ko-Rado theorem of 1961 that bounds the size of the largest such families. A recent trend has been to investigate the structure of set families with few disjoint pairs. Friedgut and Regev proved a general removal lemma, showing that when γn ≤ k ≤ ( 1 2 − γ)n, a set family with few disjoint pairs can be made intersecting by removing few sets.We provide a simple proof of a removal lemma for large families, showing that families of size close to ℓ n−1 k−1 with relatively few disjoint pairs must be close to a union of ℓ stars. Our lemma holds for a wide range of uniformities; in particular, when ℓ = 1, the result holds for all 2 ≤ k < n 2 and provides sharp quantitative estimates. We use this removal lemma to answer a question of Bollobás, Narayanan and Raigorodskii regarding the independence number of random subgraphs of the Kneser graph K(n, k). The Erdős-Ko-Rado theorem shows α(K(n, k)) = n−1 k−1 . For some constant c > 0 and k ≤ cn, we determine the sharp threshold for when this equality holds for random subgraphs of K(n, k), and provide strong bounds on the critical probability for k ≤ 1 2 (n − 3).

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“…If M is a matching in H, then we call a vertex that belongs to an edge of M an M -matched vertex. Transversals and matchings in hypergraphs have been extensively studied in the literatures (see, for example, [2,3,6,14,15,16,20,21,23] [1,5,7,10,12,19,22]).…”

Section: Introductionmentioning

confidence: 99%

“…Even for intersecting hypergraphs, a long-standing open problem known as Ryser's Conjecture is open for all r ≥ 6. Intersecting hypergraphs are well studied in the literature (see, for example, [1,5,7,10,12,19,22]).…”

Section: Introductionmentioning

confidence: 99%

Matching criticality in intersecting hypergraphs

Kang

2017

Quaestiones Mathematicae

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The transversal number τ (H) of a hypergraph H is the minimum cardinality of a set of vertices that intersects all edges of H. The matching number α (H) of H is the size of a largest matching in H, where a matching is a set of pairwise disjoint edges in H. A hypergraph is intersecting if each pair of edges has a nonempty intersection. Equivalently, H is an intersecting hypergraph if and only if α (H) = 1. We observe that τ (H) ≤ r for an intersecting hypergraph H of rank r. For an intersecting hypergraph H of rank r without isolated vertex, we call H 1-special if τ (H) = r; H is maximal 1-special if H is 1special and adding any missing r-edge to H increases the matching number. Furthermore, H is called 1-edge-critical if for any e ∈ E(H) and any v ∈ e, v-shrinking e increases the matching number; H is called 1-vertex-critical if for every vertex v in H deleting the vertex v and v-shrinking all edges incident with v increases the matching number. The intersecting hypergraphs, as defined above, are said to be matching critical in the sense that the matching number increases under above definitions of criticality [M.A. Henning and A.Yeo, Quaest. Math. 37 (2014) 127-138]. Let n i (r) (i = 2, 3, 4, 5) denote the maximum order of a hypergraph in each class of matching critical intersecting hypergraphs. In this paper we study the extremal behavior of matching critical intersecting hypergraphs. We show that n 2 (r) = n 3 (r) and n 4 (r) = n 5 (r) for all r ≥ 2, which answers an open problem on matching critical intersecting hypergraphs posed by Henning and Yeo. We also give a strengthening of the result n 4 (r) = n 5 (r) for intersecting r-uniform hypergraphs. arXiv:1512.02871v1 [math.CO] 9 Dec 2015 Lemma 3.5. For r 2, every hypergraph H in H 5 min (r) with |V (H)| = n 5 (r) is r-uniform. Proof. Suppose, to the contrary, that H contains a t-edge e such that t < r. Since H is minimal 1-vertex-critical, there exists a vertex u in e such that qd H−e (u) < 2 by Lemma 3.4, hence there exists an edge f of H whose intersecting with e is only u, i.e., e ∩ f = {u}. Clearly, f \ {u} is a transversal in H − e. Note that H has rank r, without loss of generality, we may assume that f is an r-edge of H. Let H be the hypergraph obtained from H by adding a new vertex v and a new edge (f \ {u}) ∪ {v} to H and replacing e by e ∪ {v}. By the construction of H , V (H ) = V (H) ∪ {v} and E(H ) = (E(H) \ {e}) ∪ {e ∪ {v}, (f \ {u}) ∪ {v}}. Hence |V (H )| = n 5 (r) + 1 and H has rank r. Note that f \ {u} is a transversal of H − e, so (f \ {u}) ∪ {v} meets with all edges of H . Hence H is a intersecting hypergraph with rank r. On the other hand, we note that d(v) = qd(v) = 2 and qd(u) 2 for all u ∈ H . Thus H is still 1-vertex-critical, i.e.,H ∈ H 5 (r). But then |V (H )| ≤ n 5 (r), which is a contradiction.By Lemma 3.5 and Theorem 3.1, we obtain a strengthening of Theorem 3.1.Theorem 3.2. For any integers r ≥ 2, n 4 r = n 4 (r) = n 5 (r) = n 5 r .

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Most Probably Intersecting Hypergraphs

Cited by 6 publications

References 12 publications

Domination in intersecting hypergraphs

Domination in intersecting hypergraphs

Removal and Stability for Erdös--Ko--Rado

Matching criticality in intersecting hypergraphs

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