Abu-Khazneh, A;
Barat, J;
Pokrovskiy, A;
Szabo, T;
(2019)
A family of extremal hypergraphs for Ryser's conjecture.
Journal of Combinatorial Theory, Series A
, 161
pp. 164-177.
10.1016/j.jcta.2018.07.011.
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Abstract
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
Type: | Article |
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Title: | A family of extremal hypergraphs for Ryser's conjecture |
Open access status: | An open access version is available from UCL Discovery |
DOI: | 10.1016/j.jcta.2018.07.011 |
Publisher version: | https://doi.org/10.1016/j.jcta.2018.07.011 |
Language: | English |
Additional information: | This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions. |
Keywords: | Hypergraph Matching, Ryser’s Conjecture, Extremal Structure |
UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
URI: | https://discovery-pp.ucl.ac.uk/id/eprint/10112649 |
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