What the study found
The authors prove an upper bound on the number of A-perfect matchings in uniform bipartite hypergraphs with small maximum codegree. They also derive bounds for related counting problems in Latin squares and regular hypergraphs.
Why the authors say this matters
The study suggests that these bounds help quantify how many perfect matchings, Latin square transversals, and proper edge-colorings can exist under the stated conditions. The findings indicate a way to control these counts when the maximum codegree is small.
What the researchers tested
The researchers studied bipartite hypergraphs, where every edge contains exactly one vertex from one part of the bipartition, and A-perfect matchings, which saturate every vertex in A. They proved results for uniform hypergraphs with small maximum codegree and then applied them to Latin squares and to k-uniform D-regular hypergraphs.
What worked and what didn't
The upper bound they proved applies to A-perfect matchings in the hypergraph setting they consider. Using that result, they show 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 they show 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 under specific assumptions, including uniformity, small maximum codegree, and particular conditions on n, q, and D. It does not describe limitations beyond those conditions.
Key points
- The paper proves an upper bound on A-perfect matchings in uniform bipartite hypergraphs with small maximum codegree.
- It applies this bound to show a limit on the number of transversals in some order-n Latin squares.
- The abstract states a bound of at most (n/e^{2.117})^n transversals for odd n with n ≡ 0 mod 3.
- It also bounds the number of proper q-edge-colorings in certain k-uniform D-regular hypergraphs.
- The edge-coloring bound is stated for q = (1+o(1))D and maximum codegree o(q).
Disclosure
- Research title:
- Entropy bounds limit 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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