Infinitely many counterexamples to a conjecture of Lov\'asz

TL;DR

This paper constructs an infinite family of counterexamples to Lovász's conjecture, which was a stronger statement related to matchings and vertex covers in hypergraphs, disproven for specific cases.

Contribution

It introduces a simple infinite family of counterexamples for the case r=3 and provides specific counterexamples for r=4, advancing understanding of Lovász's conjecture.

Findings

Counterexamples for r=3 based on generalized Petersen graphs

Specific counterexamples for r=4

Disproof of Lovász's conjecture in these cases

Abstract

Motivated by the well-known conjecture of Ryser which relates maximum matchings to minimum vertex covers in $r$-partite $r$-uniform hypergraphs, Lov\'asz formulated a stronger conjecture. It states that one can always reduce the matching number by removing $r-1$ vertices. This conjecture was very recently disproven for $r=3$ by Clow, Haxell, and Mohar using the line graph of a $3$-regular graph of order $102$. Building on this, we describe a simple infinite family of counterexamples based on generalized Petersen graphs for the case $r=3$ and give specific counterexamples for $r=4$.