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

Abu-Khazneh, A. and Pokrovskiy, Alexey (2017) Intersecting extremal constructions in Ryser's Conjecture for r-partite hypergraphs. Journal of Combinatorial Mathematics and Combinatorial Computing 103 , pp. 81-104. ISSN 0835-3026.

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.

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