A family of extremal hypergraphs for Ryser's conjecture

Abu-Khazneh, Ahmad; Barát, János; Pokrovskiy, Alexey; Szabó, T.

doi:10.48550/arxiv.1605.06361

Cited by 2 publications

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“…For larger values of r, truncated projective planes give an infinite family of sharp examles. Apart from these, there are some sporadic examples [4,2,8,20], moreover, Abu-Khazneh, Barát, Pokrovskiy and Szabó [1] constructed another infinite family of extremal hypergraphs but projective planes play also an important role in their construction.…”

Section: Introductionmentioning

confidence: 99%

On Ryser’s Conjecture for $t$-Intersecting and Degree-Bounded Hypergraphs

Király

Tóthmérész

2017

Electron. J. Combin.

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A famous conjecture (usually called Ryser's conjecture) that appeared in the PhD thesis of his student, J. R. Henderson [15], states that for an r-uniform r-partite hypergraph H, the inequality τ (H) ≤ (r − 1)·ν(H) always holds.This conjecture is widely open, except in the case of r = 2, when it is equivalent to Kőnig's theorem [18], and in the case of r = 3, which was proved by Aharoni in 2001 [3].Here we study some special cases of Ryser's conjecture. First of all, the most studied special case is when H is intersecting. Even for this special case, not too much is known: this conjecture is proved only for r ≤ 5 in [10,21]. For r > 5 it is also widely open.Generalizing the conjecture for intersecting hypergraphs, we conjecture the following. If an r-uniform r-partite hypergraph H is t-intersecting (i.e., every two hyperedges meet in at least t < r vertices), then τ (H) ≤ r − t. We prove this conjecture for the case t > r/4. Gyárfás [10] showed that Ryser's conjecture for intersecting hypergraphs is equivalent to saying that the vertices of an r-edge-colored complete graph can be covered by r − 1 monochromatic components.Motivated by this formulation, we examine what fraction of the vertices can be covered by r − 1 monochromatic components of different colors in an r-edgecolored complete graph. We prove a sharp bound for this problem.Finally we prove Ryser's conjecture for the very special case when the maximum degree of the hypergraph is two.

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Section: Introductionmentioning

confidence: 99%

On Ryser’s Conjecture for $t$-Intersecting and Degree-Bounded Hypergraphs

Király

Tóthmérész

2017

Electron. J. Combin.

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“…As in most known constructions for this problem (for example [2]), our construction will be based on finite projective planes and the corresponding affine planes. Recall that a projective plane of order p is a (p + 1)-uniform hypergraph with p 2 + p + 1 vertices, with the property that each pair of edges (called lines) intersects in exactly one vertex, and each pair of vertices (called points) is contained in exactly one line.…”

Section: Basic Constructionmentioning

confidence: 99%

“…Ryser's Conjecture is tight for a given value of r if there is an r-partite runiform hypergraph H with τ (H) ≥ (r−1)ν(H) (such hypergraphs are called r-Ryser hypergraphs in [2]). Because of the apparent difficulty of the problem in general, a significant amount of work has been done on constructing and understanding r-Ryser hypergraphs (e.g.…”

Section: Introductionmentioning

confidence: 99%

“…For every prime power p there is a standard construction, based on the projective plane, that gives an intersecting r-Ryser hypergraph for r = p + 1. Very recently it was proved in [2] that intersecting r-Ryser hypergraphs exist also for every r = p + 2. Apart from these infinite families, the only other values of r for which the conjecture is known to be tight are r = 7 [4,1], r = 11 [1] and r = 12 [6].…”

Section: Introductionmentioning

confidence: 99%

“…Apart from these infinite families, the only other values of r for which the conjecture is known to be tight are r = 7 [4,1], r = 11 [1] and r = 12 [6]. In [2] the authors ask whether there exists a constant K such that for every r there exists an intersecting r-partite r-uniform hypergraph H with τ (H) ≥ r − K, thus showing that Ryser's problem is close to being best possible for every r. Here we answer this question in the affirmative, by showing in particular that we may take K ≤ 4 for all sufficiently large integers r. This is our main result (Theorem 6), and it appears in Section 4. We also give a new construction for intersecting r-Ryser hypergraphs for special values of r in Section 5.…”

Section: Introductionmentioning

confidence: 99%

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