A local spectral condition for perfect matchings in 3-graphs

TL;DR

This paper establishes spectral conditions on local neighborhoods in 3-uniform hypergraphs that guarantee the existence of perfect or fractional matchings, extending and tightening previous degree-based results.

Contribution

It introduces new spectral bounds based on the spectral radius of local 2-graph neighborhoods that ensure perfect and fractional matchings in 3-graphs, improving upon prior degree conditions.

Findings

Spectral radius conditions guarantee perfect matchings in large 3-graphs.

Derived tight bounds for fractional matchings based on spectral radius.

Extended classical degree-based matching results to spectral conditions.

Abstract

Let $\gamma$ be a constant such that $0 < \gamma < 1$, and let $n$ be a sufficiently large integer. Consider a $3$-uniform hypergraph $H$ on $n$ vertices. In 2013, K\"{u}hn, Osthus, and Treglown, along with Khan independently, proved that for large enough $n$ with $n\equiv 0\pmod{3}$, if $\delta_1(H)\geq\binom{2n/3}{2}$, 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 $\rho(N_H(v)) > (2/3+\gamma)n$ for all $v\in V(H)$, where $\rho(N_H(v))$ denotes the spectral radius of $N_H(v)$, then $H$ has a perfect matching. This bound is asymptotically tight. Furthermore, for integer $s$ satisfying $n\geq 3s+3$, we establish that if \[ \rho(N_H(v))>\frac{1}{2}(s-1+\sqrt{(s-1)^2+4s(n-s-1)})\] holds for every $v\in V(H),$ then $H$ admits a fractional matching of size $s+1$. Notably, this second spectral bound is tight.