Intersecting extremal constructions in Ryser's Conjecture for rpartite hypergraphs
AbuKhazneh, A. and Pokrovskiy, Alexey (2017) Intersecting extremal constructions in Ryser's Conjecture for rpartite hypergraphs. Journal of Combinatorial Mathematics and Combinatorial Computing 103 , pp. 81104. ISSN 08353026.

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Abstract
Ryser's Conjecture states that for any rpartite runiform 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 rpartite 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 rpartite 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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Item Type:  Article 

School:  Birkbeck Faculties and Schools > Faculty of Business and Law > Birkbeck Business School 
Depositing User:  Alexey Pokrovskiy 
Date Deposited:  21 Jan 2019 10:35 
Last Modified:  02 Aug 2023 17:47 
URI:  https://eprints.bbk.ac.uk/id/eprint/25887 
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