A family of extremal hypergraphs for Ryser's conjecture

• Published in: Journal of combinatorial theory. Series A 2019-01, Vol.161, p.164-177
• Main Authors: Abu-Khazneh, A., Barát, János, Pokrovskiy, A., Szabó, Tibor
• Format: Article
• Language: English
• Subjects: Extremal structure; Hypergraph matching; Ryser's conjecture
• ISSN: 0097-3165; 1096-0899
• DOI: 10.1016/j.jcta.2018.07.011

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.