Local spectral matching conjecture for 3-graphs

Let $N_H(v)$ denote the link graph of a vertex $v$ in a $3$-graph $H$, and let $\rho(N_H(v))$ be its spectral radius. For integers $n,s$ with $n\geq 3s+3$, let $H$ be a $3$-graph with vertex set $[n]$.

Local spectral matching conjecture. If, for every $v\in V(H)$,

$$\rho\big(N_H(v)\big)>\frac{1}{2}\left(s-1+\sqrt{(s-1)^2+4s(n-s-1)}\right),$$

then $H$ has a matching of size $s+1$. Moreover, if $n=3s+3$ and, for every $v\in V(H)$,

$$\rho\big(N_H(v)\big)>2n/3-2,$$

then $H$ contains a perfect matching.

This conjecture asks whether a lower bound on the spectral radius of every link graph forces a large, or in the extremal case perfect, matching in a $3$-graph. The preceding theorem establishes a related sufficient condition with threshold $(2/3+\gamma)n$ for sufficiently large $n$ divisible by $3$; the conjectured bounds are presented as asymptotically tight based on the stated constructions, and the supplied text gives no proof of the conjecture itself.