A spectral condition for perfect matchings in 3-partite 3-graphs

Abstract

Let $H$ be a 3-partite 3-uniform hypergraph whose three vertex classes all have size $n$. For a vertex $v \in V(H)$, the link graph $N_H(v)$ is defined on $V(H)\setminus\{v\}$ with edge set $\{e\setminus\{v\}: v\in e\in E(H)\}$, and we denote by $ρ(N_H(v))$ its spectral radius. We prove that for every $α>0$ there exists $n_0$ such that for all $n\ge n_0$ the following holds: if \[ ρ\bigl(N_H(v)\bigr) > \left(\frac{\sqrt{2}}{2}+α\right)n \] for every vertex $v\in V(H)$, then $H$ contains a perfect matching. This spectral condition is asymptotically best possible.