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arXiv · 1809.06931

Non-intersecting Ryser hypergraphs

Anurag Bishnoi·Valentina Pepe
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Abstract

A famous conjecture of Ryser states that every $r$-partite hypergraph has vertex cover number at most $r - 1$ times the matching number. In recent years, hypergraphs meeting this conjectured bound, known as $r$-Ryser hypergraphs, have been studied extensively. It was recently proved by Haxell, Narins and Szabó that all $3$-Ryser hypergraphs with matching number $ν> 1$ are essentially obtained by taking $ν$ disjoint copies of intersecting $3$-Ryser hypergraphs. Abu-Khazneh showed that such a characterisation is false for $r = 4$ by giving a computer generated example of a $4$-Ryser hypergraph with $ν= 2$ whose vertex set cannot be partitioned into two sets such that we have an intersecting $4$-Ryser hypergraph on each of these parts. Here we construct new infinite families of $r$-Ryser hypergraphs, for any given matching number $ν> 1$, that do not contain two vertex disjoint intersecting $r$-Ryser subhypergraphs.

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Anurag Bishnoi, Valentina Pepe. 2019-10-28. Non-intersecting Ryser hypergraphs. https://arxiv.org/abs/1809.06931

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