Exact local spectral thresholds for perfect matchings in $3$-graphs and $3$-partite $3$-graphs

Abstract

For a $3$-uniform hypergraph $H$, let $\sigma(H)$ be the minimum over the vertices of $H$ of the spectral radius of the link. Lin, Lu, Yuan and Zhao conjectured that $\sigma(H)>\tfrac{2n}3-2$ forces a perfect matching in a $3$-graph of order $n$ divisible by three, and Lu and Yuan conjectured that $\sigma(H)>\tau(q)$ forces one in a $q$-balanced $3$-partite $3$-graph, where $\tau(q)=\sqrt{q(q-1)/2}$ for odd $q$ and $\tau(q)$ is given by a quartic for even $q$. In this paper, we prove both perfect matching conjectures for large order. Each proof passes through a stability theorem for fractional matchings: a fractional vertex cover of deficient weight with a zero coordinate is obtained, the link of that vertex carries the induced cover, and the spectral radius of such a link is bounded by an inequality whose extremal cases identify the corresponding space barriers.