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arXiv · 2505.05339

A Counterexample to a Conjecture of Lov\'asz

Alexander Clow·Penny Haxell·Bojan Mohar
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Abstract

In 1975 Lov\'{a}sz conjectured that every $r$-partite, $r$-uniform hypergraph contains $r-1$ vertices whose deletion reduces the matching number. If true, this statement would imply a well-known conjecture of Ryser from 1971, which states that every $r$-partite, $r$-uniform hypergraph has a vertex cover of size at most $r-1$ times its matching number. When $r=2$, Ryser's conjecture is simply K\H{o}nig's theorem, and the conjecture of Lov\'asz is an immediate corollary. Ryser's conjecture for $r=3$ was proven by Aharoni in 2001, and remains open for all $r\geq 4$. Here we show that the conjecture of Lov\'asz is false in the case $r=3$. Our counterexample is the line hypergraph of the Biggs-Smith graph, a highly symmetric cubic graph on 102 vertices.

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Alexander Clow, Penny Haxell, Bojan Mohar. 2025-05-08. A Counterexample to a Conjecture of Lov\'asz. https://arxiv.org/abs/2505.05339

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