A Counterexample to a Conjecture of Lov\'asz

TL;DR

This paper presents a counterexample to Lovász's conjecture for 3-uniform hypergraphs, disproving its validity in that case and impacting related conjectures in hypergraph theory.

Contribution

The paper constructs a specific counterexample using the line hypergraph of the Biggs-Smith graph, showing Lovász's conjecture is false for r=3.

Findings

Counterexample disproves Lovász's conjecture for r=3

Lovász's conjecture does not hold universally for all r

Implications for related hypergraph conjectures

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.