What the study found
The authors prove an upper bound on the number of A-perfect matchings in uniform hypergraphs with small maximum codegree, where a hypergraph is bipartite if each edge contains exactly one vertex from one part of the vertex set and an A-perfect matching covers every vertex in A. They also derive bounds for transversals in certain Latin squares and for proper edge-colorings of regular hypergraphs.
Why the authors say this matters
The abstract does not give a broader practical motivation beyond these counting bounds. The findings indicate that the authors' results apply to several related combinatorial settings, including Latin squares and edge-colorings.
What the researchers tested
The researchers studied uniform hypergraphs under a small maximum codegree condition and counted A-perfect matchings. They then applied the result to order-n Latin squares and to k-uniform D-regular hypergraphs, considering the case where q is approximately D and the maximum codegree is o(q).
What worked and what didn't
They proved an upper bound for A-perfect matchings in the hypergraph setting described in the abstract. Using that bound, they showed that there exist order-n Latin squares with at most (n/e^{2.117})^n transversals when n is odd and n ≡ 0 mod 3, and that k-uniform D-regular hypergraphs on n vertices have at most ((1+o(1))q/e^k)^{Dn/k} proper q-edge-colorings when q = (1+o(1))D and the maximum codegree is o(q).
What to keep in mind
The abstract only states results for uniform hypergraphs with small maximum codegree, specific Latin-square orders, and the edge-coloring regime where q is close to D. It does not describe proof details, limitations beyond these conditions, or any empirical validation.
Key points
- An upper bound was proved for the number of A-perfect matchings in uniform bipartite hypergraphs with small maximum codegree.
- The paper gives a bound on transversals in certain order-n Latin squares when n is odd and n ≡ 0 mod 3.
- The authors also bound the number of proper q-edge-colorings in k-uniform D-regular hypergraphs when q is approximately D.
- The abstract defines a bipartite hypergraph as one where each edge has exactly one vertex in part A.
- The abstract does not describe practical applications or proof details.
Disclosure
- Research title:
- Upper bounds found for perfect matchings in bipartite hypergraphs
- Authors:
- Tantan Dai, Alexander Divoux, Tom Kelly
- Institutions:
- Georgia Institute of Technology, Georgia Institute of Technology, Princeton University
- Publication date:
- 2026-04-23
- DOI:
- 10.37236/14339
- OpenAlex record:
- View
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