Extremal hypergraphs for Ryser's Conjecture

Haxell, Penny; Narins, Lothar; Szabó, Tibor

doi:10.1016/j.jcta.2018.04.004

Cited by 10 publications

(14 citation statements)

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“…Moreover, the only other known family of extremals [1] is also constructed using truncated projective planes. In [12] it is shown that the truncated Fano-plane is the main building block in the characterization of the sharp hypergraphs for Ryser's conjecture in the case r = 3. In addition, the near-extremal family recently constructed by Haxell and Scott [13] is also based on truncated projective planes.…”

Section: Recall the Definition Of A Truncated Projective Planementioning

confidence: 99%

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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: Recall the Definition Of A Truncated Projective Planementioning

confidence: 99%

“…The hypergraphs achieving τ (H) = (r − 1) · ν(H) have also been investigated, but this problem is also widely open. Haxell, Narins and Szabó characterized the sharp examples for r = 3 [11,12]. For larger values of r, truncated projective planes give an infinite family of sharp examles.…”

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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“…The 2-Ryser poset R 2 is an infinite chain of stars with K 2 as its minimal element. The poset R 3 was determined in [9] (for arbitrary matching numbers). In the intersecting case, R 3 has a unique minimal element: the 3-graph R obtained from T 3 by deleting one of its edges.…”

Section: The Ryser Posetmentioning

confidence: 99%

A family of extremal hypergraphs for Ryser's conjecture

Abu-Khazneh

Barát

Pokrovskiy

et al. 2019

Journal of Combinatorial Theory, Series A

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Ryser's Conjecture states that for any r-partite r-uniform hypergraph, the vertex cover number is at most r−1 times the matching number. This conjecture is only known to be true for r ≤ 3 in general and for r ≤ 5 if the hypergraph is intersecting. There has also been considerable effort made for finding hypergraphs that are extremal for Ryser's Conjecture, i.e. r-partite hypergraphs whose cover number is r − 1 times its matching number. Aside from a few sporadic examples, the set of uniformities r for which Ryser's Conjecture is known to be tight is limited to those integers for which a projective plane of order r − 1 exists.We produce a new infinite family of r-uniform hypergraphs extremal to Ryser's Conjecture, which exists whenever a projective plane of order r − 2 exists. Our construction is flexible enough to produce a large number of non-isomorphic extremal hypergraphs. In particular, we define what we call the Ryser poset of extremal intersecting r-partite r-uniform hypergraphs and show that the number of maximal and minimal elements is exponential in √ r.This provides further evidence for the difficulty of Ryser's Conjecture.

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“…As mentioned before, the existence of these intersecting r-Ryser hypergraphs also implies the existence of r-Ryser hypergraphs with arbitrary matching number, by simply taking disjoint copies. Moreover, it was proved by Haxell, Narins and Szabó [12] that for r = 3 every r-Ryser hypergraph H contains ν(H) many disjoint copies of intersecting r-Ryser hypergraphs. Abu-Khazneh obtained computer generated examples which showed that such a result cannot hold true for r = 4 [1, Ch.5], and in [2] it was asked how the characterisation of the 3-Ryser hypergraphs can be generalised to higher values of r. Here, we construct non-intersecting r-Ryser hypergraphs which prove that this characterisation fails for infinitely many values of r. Our first construction works whenever r − 1 is an odd prime, while our second construction works whenever r − 1 is any prime power greater than or equal to 4.…”

Section: Introductionmentioning

confidence: 99%

Nonintersecting Ryser Hypergraphs

Bishnoi¹,

Pepe

2020

SIAM J. Discrete Math.

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A famous conjecture of Ryser states that every r-partite hypergraph has vertex cover number at most r − 1 times the matching number. In recent years, hypergraphs meeting this conjectured bound, known as r-Ryser hypergraphs, have been studied extensively. It was recently proved by Haxell, Narins and Szabó that all 3-Ryser hypergraphs with matching number ν > 1 are essentially obtained by taking ν disjoint copies of intersecting 3-Ryser hypergraphs. Abu-Khazneh showed that such a characterisation is false for r = 4 by giving a computer generated example of a 4-Ryser hypergraph with ν = 2 whose vertex set cannot be partitioned into two sets such that we have an intersecting 4-Ryser hypergraph on each of these parts.Here we construct new infinite families of r-Ryser hypergraphs, for any given matching number ν > 1, that do not contain two vertex disjoint intersecting r-Ryser subhypergraphs.

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Extremal hypergraphs for Ryser's Conjecture

Cited by 10 publications

References 11 publications

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

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

A family of extremal hypergraphs for Ryser's conjecture

Nonintersecting Ryser Hypergraphs

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