Non-intersecting Ryser hypergraphs

TL;DR

This paper constructs infinite families of $r$-Ryser hypergraphs with a given matching number that cannot be decomposed into intersecting subhypergraphs, challenging previous characterizations for higher uniformities.

Contribution

It introduces new infinite families of $r$-Ryser hypergraphs that lack the intersecting subhypergraph decomposition, extending understanding beyond the case $r=3$.

Findings

Constructed infinite families of $r$-Ryser hypergraphs for any $ u > 1$

Demonstrated these hypergraphs do not contain two vertex disjoint intersecting $r$-Ryser subhypergraphs

Challenged previous characterizations for $r eq 3$

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 Szab\'{o} that all $3$-Ryser hypergraphs with matching number $\nu > 1$ are essentially obtained by taking $\nu$ 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 $\nu = 2$ 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 $\nu > 1$, that do not contain two vertex disjoint intersecting $r$-Ryser subhypergraphs.