Intersecting extremal constructions in Ryser's Conjecture for r-partite hypergraphs

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. For intersecting hypergraphs, Ryser's Conjecture reduces to saying that the edges of every r-partite intersecting hypergraph can be covered by r − 1 vertices. This special case of the conjecture has only been proven for r ≤ 5.It is interesting to study hypergraphs which are extremal in Ryser's Conjecture i.e, those hypergraphs for which the vertex cover number is exactly r − 1 times the matching number. There are very few known constructions of such graphs. For large r the only known constructions come from projective planes and exist only when r − 1 is a prime power. Mansour, Song and Yuster studied how few edges a hypergraph which is extremal for Ryser's Conjecture can have. They defined f (r) as the minimum integer so that there exist an r-partite intersecting hypergraph H with τ (H) = r − 1 and with f (r) edges. They showed that f (3) = 3, f (4) = 6, f (5) = 9, and 12 ≤ f (6) ≤ 15.In this paper we focus on the cases when r = 6 and 7. We show that f (6) = 13 improving previous bounds. We also show that f (7) ≤ 22, giving the first known extremal hypergraphs for the r = 7 case of Ryser's Conjecture. These results have been obtained independently by Aharoni, Barat, and Wanless.

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

“…For larger values of r, truncated projective planes give an infinite family of sharp examles. Apart from these, there are some sporadic examples [4,2,8,20], moreover, Abu-Khazneh, Barát, Pokrovskiy and Szabó [1] constructed another infinite family of extremal hypergraphs but projective planes play also an important role in their construction.…”

Covers in partitioned intersecting hypergraphs

“…This is known to be sharp for r for which there exists an r-uniform projective plane. In [1,3] the conjecture was shown to be sharp also for the first value of r for which an r-uniform projective plane does not exist, namely r = 7. It is plausible that the conjecture is sharp for all r.…”