A local spectral condition for perfect matchings in 3-graphs

Abstract

Let ${\hat{I}}^3$ be a constant such that $0<{\hat{I}}^3<1$, and let $n$ be a sufficiently large integer. Consider a $3$-uniform hypergraph $H$ on $n$ vertices. In 2013, Kühn, Osthus, and Treglown, along with Khan independently, proved that for large enough $n$ with $n\equiv 0(\mathrm{mod}3)$, if $\hat{I}{´}_1(H)\ge \left(\genfrac{}{}{0ex}{}{2n/3}{2}\right)$, then $H$ admits a perfect matching. For any vertex $v\in V(H)$, we define $N_H(v)$ as the $2$-graph with vertex set $V(H)\setminus \{v\}$ and edge set $E(N_H(v))=\{e\subseteq V(H)\setminus \{v\}:e\cup \{v\}\in E(H)\}$. In this paper, we show that if $\ddot{I}(N_H(v))>(2/3+{\hat{I}}^3)n$ for all $v\in V(H)$, where $\ddot{I}(N_H(v))$ denotes the spectral radius of $N_H(v)$, then $H$ has a perfect matching. This bound is asymptotically tight. Furthermore, for integer $s$ satisfying $n\ge 3s+3$, we establish that if \[ ρ(N_H(v))>\frac{1}{2}(s-1+\sqrt{(s-1)^2+4s(n-s-1)})\] holds for every $v\in V(H),$ then $H$ admits a fractional matching of size $s+1$. Notably, this second spectral bound is tight.

16 pages