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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2506.21286 |
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| _version_ | 1866909689198936064 |
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| author | Abiad, Aida Garbe, Frederik Povill, Xavier Spiegel, Christoph |
| author_facet | Abiad, Aida Garbe, Frederik Povill, Xavier Spiegel, Christoph |
| contents | 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 $r=4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_21286 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Infinitely many counterexamples to a conjecture of Lovász Abiad, Aida Garbe, Frederik Povill, Xavier Spiegel, Christoph Combinatorics 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 $r=4$. |
| title | Infinitely many counterexamples to a conjecture of Lovász |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2506.21286 |