Ryser–Brualdi–Stein conjecture

Prove that every decomposition of the edges of the complete bipartite graph K_{n,n} into n edge-disjoint matchings of size n contains a rainbow matching of size n−1 when n is even and of size n when n is odd.

Background

The Ryser–Brualdi–Stein conjecture concerns rainbow matchings in decompositions of the complete bipartite graph into n edge-disjoint perfect matchings. The conjectured rainbow matching size is n−1 for even n and n for odd n, with the n−1 bound in the even case shown to be best possible by constructions derived from Cayley tables of suitable abelian groups.

The paper notes that the conjecture has received substantial attention and that Montgomery proved the n−1 lower bound for sufficiently large even n. The exact general conjecture is nevertheless explicitly presented as a conjecture in the paper.

References

The famous Ryser-Brualdi-Stein Conjecture claims that every decomposition of the edges of the complete bipartite graph Kn,n into n edge-disjoint matchings of size n, M1, . . . , Mn, admits a rainbow matching M ⊂ ⋃i∈[n] Mi of size n − 1 when n is even, and a rainbow matching of size n when n is odd.

— A note on improved bounds for hypergraph rainbow matching problems  (2501.03216 - Bowtell et al., 6 Jan 2025) in Section 1, Introduction, page 1

The natural extremal problem on the size of the largest partial transversal that always exists is the topic of the well-known Ryser-Brualdi-Stein conjecture, with origins from 1967, which suggests that every Latin square of order $n$ should have a transversal when $n$ is odd, and a partial transversal with $n-1$ cells when $n$ is even.

— Almost every Latin square has a decomposition into transversals  (2501.05438 - Bowtell et al., 9 Jan 2025) in Section 1, Introduction