On Ryser’s conjecture for linear intersecting multipartite hypergraphs

Francetić, Nevena; Herke, Sarada; McKay, Brendan D.; Wanless, Ian M.

doi:10.1016/j.ejc.2016.10.004

Cited by 12 publications

(24 citation statements)

References 8 publications

(58 reference statements)

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“…The following strengthening of Ryser's conjecture was phrased by Aharoni et al [4,Conjecture 3.1]: "In an intersecting r-partite r-uniform hypergraph H, there exists a class of size r − 1 or less, or a cover of the form e − {x} for some e ∈ E and x ∈ e." This conjecture was disproved in [8]. Note however, that by Theorem 15, if we require the cover to be multi-colored, then additionally requiring it to be a subset of a hyperedge does not decrease the number of coverable hyperedges in the worst case.…”

Section: Covering Large Fraction By Few Monochromatic Componentsmentioning

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: Covering Large Fraction By Few Monochromatic Componentsmentioning

confidence: 99%

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

Király

Tóthmérész

2017

Electron. J. Combin.

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“…Except for these two cases, the conjecture is wide open. If we assume that the hypergraph is also intersecting, then it is known to be true for r ≤ 5, as proved by Tuza [19], and if we further assume that the hyeprgraph is linear, that is, every two edges intersect in a unique vertex, then Francetić, Herke, McKay, and Wanless [11] have proved it for r ≤ 9.…”

Section: Introductionmentioning

confidence: 90%

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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“…However, the conjecture is also known to be true for some special cases. In particular, it has been proven by Tuza [15] for r-partite intersecting hypergraphs when r ≤ 5, and by Francetić, Herke, McKay, and Wanless [8] for r ≤ 9, when one makes the further assumption that any two edges of the r-partite hypergraph intersect in exactly one vertex.…”

Section: Introductionmentioning

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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On Ryser’s conjecture for linear intersecting multipartite hypergraphs

Cited by 12 publications

References 8 publications

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

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

Nonintersecting Ryser Hypergraphs

A family of extremal hypergraphs for Ryser's conjecture

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