# Non-intersecting Ryser hypergraphs

@article{Bishnoi2018NonintersectingRH, title={Non-intersecting Ryser hypergraphs}, author={A. Bishnoi and V. Pepe}, journal={arXiv: Combinatorics}, year={2018} }

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 Szab\'{o} that all $3$-Ryser hypergraphs with matching number $\nu > 1$ are essentially obtained by taking $\nu$ disjoint copies of intersecting $3$-Ryser hypergraphs. Abu-Khazneh showed that… Expand

#### References

SHOWING 1-10 OF 21 REFERENCES

A family of extremal hypergraphs for Ryser's conjecture

- Mathematics, Computer Science
- J. Comb. Theory, Ser. A
- 2019

The Ryser poset of extremal intersecting r-partite and r-uniform hypergraphs is defined and it is shown that it has exponentially many maximal and minimal elements, providing further evidence for the difficulty of RYSer's Conjecture. Expand

Multipartite Hypergraphs Achieving Equality in Ryser’s Conjecture

- Mathematics, Computer Science
- Graphs Comb.
- 2016

A fractional version of the following stronger form of Ryser’s conjecture is proved: in an r-partite hypergraph there exists a set of size at most r-1, contained either in one side of the hypergraph or in an edge, whose removal reduces the matching number by 1. Expand

On Ryser's conjecture for linear intersecting multipartite hypergraphs

- Mathematics, Computer Science
- Eur. J. Comb.
- 2017

It is found that r = 8 is the smallest value of r for which there exists a linear intersecting r -partite hypergraph that achieves ź = r - 1 and is not isomorphic to a subhypergraph of a projective plane. Expand

Extremal hypergraphs for Ryser's Conjecture

- Mathematics, Computer Science
- J. Comb. Theory, Ser. A
- 2018

The goal in this paper is to characterize those hypergraphs which are tight for Aharoni's Theorem and the characterization proves an old and wide open strengthening of Ryser's Conjecture for the 3-uniform extremal case, that is, forhypergraphs with τ = 2 ν . Expand

Matchings and covers of multipartite hypergraphs

- Mathematics
- 2016

Koenig’s theorem is a classic result in combinatorics which states that for every bipartite graph G, the cover number of G (denoted by τ (G)) is equal to its matching number (denoted by ν(G)). The… Expand

A Note on Intersecting Hypergraphs with Large Cover Number

- Mathematics, Computer Science
- Electron. J. Comb.
- 2017

This work gives a construction of r-partite r-uniform intersecting hypergraphs with cover number at least r-4 for all but finitely many r and shows that a long-standing unsolved conjecture due to Ryser is close to being best possible for every value of r. Expand

What did Ryser Conjecture

- Mathematics
- 2018

Two prominent conjectures by Herbert J. Ryser have been falsely attributed to a somewhat obscure conference proceedings that he wrote in German. Here we provide a translation of that paper and try to… Expand

Maximal intersecting families and affine regular polygons in PG(2, q)

- Mathematics, Computer Science
- J. Comb. Theory, Ser. A
- 1989

A new upper bound for m(k), the minimum number of members of a maximal k-clique, is given, proving m( k) ⩽ k 2 2 + 5k + o(k) whenever k − 1 is a rime power. Expand

Permutation decomposition of (0,1)-matrices and decomposition transversals

- Mathematics
- 1971

The central problem of this thesis is the study of sums of disjoint partial permutation matrices ("permutation decompositions"). This problem has as its origin the result of G. Birkoff that an ordern… Expand

On the size of a blocking set inPG(2,p)

- Mathematics, Computer Science
- Comb.
- 1994

We show that the size of a non-trivial blocking set in the Desarguesian projective planePG(2,p), wherep is prime, is at least 3(p+1)/2. This settles a 25 year old conjecture of J. di Paola.