A local spectral condition for perfect matchings in 3-graphs

Abstract: Let $γ$ be a constant such that $0 < γ< 1$, 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 $n\equiv 0\pmod{3}$, if $δ_1(H)\geq\binom{2n/3}{2}$, 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 $ρ(N_H(v)) &gt; (2/3+γ)n$ for all $v\in V(H)$, where $ρ(N_H(v))$ 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 [ 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.

• The paper introduces a local spectral criterion by applying the spectral radius of vertex link graphs to guarantee perfect matchings in 3-uniform hypergraphs.
• It establishes explicit, asymptotically tight spectral thresholds for both perfect and fractional matchings using advanced spectral theory and probabilistic methods.
• The work provides constructive extremal examples and paves the way for utilizing spectral conditions in combinatorial optimization and algorithm design.

Local Spectral Conditions for Perfect Matchings in 3-Uniform Hypergraphs

Background and Motivation

The combinatorial study of matchings in hypergraphs, especially $3$-uniform hypergraphs (3-graphs), occupies a central place in extremal set theory and combinatorial optimization. The existence of perfect matchings in $3$-graphs under degree conditions has been thoroughly investigated, culminating in tight results for minimum vertex degree thresholds. Traditional approaches rely on combinatorial or probabilistic arguments related to the minimum $1$-degree ($\delta_1(H)$). This paper introduces a spectral perspective, focusing on the spectral radius $\rho(N_H(v))$ of the link graph associated with each vertex $v$, offering a novel local spectral criterion for guaranteeing perfect matchings.

Main Results and Theoretical Contributions

The paper establishes that a sufficiently large spectral radius for the link graph of each vertex is a robust condition for the existence of perfect matchings in $3$-graphs. The principal theorem asserts that for sufficiently large $n$ (with $n \equiv 0 \pmod{3}$) and $0 < \gamma \ll 1$, if

$3$0

for all $3$1, then $3$2 admits a perfect matching. The authors demonstrate that this bound is asymptotically tight by explicitly constructing extremal examples where the bound is met but no perfect matching exists.

Further, for integers $3$3 with $3$4, they prove that if

$3$5

holds for every $3$6, then $3$7 has a fractional matching of size $3$8. This bound is proved to be tight, with constructive extremal examples provided.

A conjecture extending these results to actual matchings (as opposed to fractional matchings) is stated, proposing that the same spectral bound ensures existence of matchings of size $3$9, and, in the extremal case $1$0, guarantees a perfect matching.

Three main theorems encapsulate these contributions:

• If the local spectral radius exceeds $1$1, then a perfect matching exists (asymptotically tight).
• For $1$2, exceeding the derived spectral threshold implies a matching of size $1$3.
• For $1$4, the spectral threshold guarantees a fractional matching of size $1$5, and for $1$6 and $1$7, a perfect fractional matching exists.

Proof Methods and Structural Ingredients

The proofs employ advanced spectral graph theory (notably tight spectral radius bounds in terms of matching number and graph size), probabilistic methods (Chernoff bounds and nibble lemma for hypergraphs), and an absorbing lemma tailored for hypergraphs with large link spectral radii. Key ingredients include:

• Stanley's, Hong-Shu-Fang’s, and Nikiforov's sharp spectral bounds relating edge count, minimum degree, and spectral radius.
• Nordhaus–Gaddum-type inequalities for spectral radii to estimate the number of common edges in link graphs.
• Spectral extremal characterizations from Feng-Yu-Zhang [FYZ17].
• Absorbing structures and probabilistic construction techniques for greedy matching extension, utilizing fractional matching duality and rainbow fractional matchings.

The absorbing lemma guarantees the existence of small absorbing matchings capable of extending almost perfect matchings to true perfect matchings, under the local spectral conditions. The proofs further leverage fractional vertex covers and linear programming duality to transition fractional matching existence into actual matching sizes.

Numerical Bounds and Tightness

The numerical thresholds are explicit and sharp:

• The perfect matching threshold $1$8 for the spectral radius is asymptotically tight, demonstrated via extremal construction.
• The fractional matching threshold using $1$9 is proved to be tight both theoretically and via explicit construction.

These results extend, complement, and refine the existing minimum degree thresholds by translating them into the spectral domain, enabling local analysis via spectral graph invariants.

Implications and Future Directions

The shift from degree conditions to spectral radius conditions introduces a powerful local criterion for perfect matchings in $\delta_1(H)$0-graphs. Practically, spectral radius can be computed efficiently for large graphs and captures richer local structural information than degree alone. The findings have implications for relaxation-based algorithms in combinatorial optimization, random hypergraph models — where spectral properties govern phase transitions — and offer new perspectives in spectral extremal combinatorics.

The conjecture proposing that matching existence can be assured at the same spectral threshold as fractional matchings is particularly notable. If proven, it would establish a seamless transition between integer and fractional matchings in spectral terms. Further research may extend these results to higher uniformity hypergraphs ($\delta_1(H)$1-graphs with $\delta_1(H)$2), investigate analogous spectral thresholds for other combinatorial structures (e.g., Hamilton cycles), and integrate local spectral conditions into algorithmic matching heuristics.

Conclusion

This paper rigorously establishes asymptotically tight local spectral criteria for the existence of perfect and fractional matchings in $\delta_1(H)$3-uniform hypergraphs (2604.13726). The utilization of the spectral radius of vertex link graphs enables finer, local structural guarantees than traditional degree-based approaches. The results are underpinned by sharp theoretical bounds and constructive extremal examples. The implications pave the way for further exploration into spectral conditions for combinatorial optimization in hypergraphs and potentially broader classes of discrete structures.