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arXiv · 2506.21286

Infinitely many counterexamples to a conjecture of Lovász

Aida Abiad·Frederik Garbe·Xavier Povill·Christoph Spiegel
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Abstract

Motivated by the well-known conjecture of Ryser which relates maximum matchings to minimum vertex covers in $r$-partite $r$-uniform hypergraphs, Lovász 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 $r=4$.

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Aida Abiad, Frederik Garbe, Xavier Povill, Christoph Spiegel. 2025-07-14. Infinitely many counterexamples to a conjecture of Lovász. https://arxiv.org/abs/2506.21286

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