Exact local spectral thresholds for perfect matchings in $3$-graphs and $3$-partite $3$-graphs

@misc{liu2026exact,
  title = {{Exact local spectral thresholds for perfect matchings in $3$-graphs and $3$-partite $3$-graphs}},
  author = {Pei Liu and Suil O},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.26832},
  eprint = {2609.26832},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.26832},
}

TY  - UNPB
TI  - Exact local spectral thresholds for perfect matchings in $3$-graphs and $3$-partite $3$-graphs
AU  - Pei Liu
AU  - Suil O
PY  - 2026
DA  - 2026/09/21
PB  - arXiv
DO  - 10.48550/arxiv.2609.26832
UR  - https://arxiv.org/abs/2609.26832
AB  - For a $3$-uniform hypergraph H, let σ(H) be the minimum over the vertices of H of the spectral radius of the link. Lin, Lu, Yuan and Zhao conjectured that σ(H)>2n/3−2 forces a perfect matching in a $3$-graph of order n divisible by three, and Lu and Yuan conjectured that σ(H)>τ(q) forces one in a q-balanced $3$-partite $3$-graph, where τ(q)=√(q(q−1)/2) for odd q and τ(q) is given by a quartic for even q. In this paper, we prove both perfect matching conjectures for large order. Each proof passes through a stability theorem for fractional matchings: a fractional vertex cover of deficient weight with a zero coordinate is obtained, the link of that vertex carries the induced cover, and the spectral radius of such a link is bounded by an inequality whose extremal cases identify the corresponding space barriers.
ER  -