A Counterexample to a Conjecture of Lovász

Abstract

In 1975 Lovász 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 Kőnig's theorem, and the conjecture of Lovász is an immediate corollary. Ryser's conjecture for $r=3$ was proven by Aharoni in 2001, and remains open for all $r\ge 4$. Here we show that the conjecture of Lovász 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.

23 pages, 4 figures, 2 tables