2016/05/18 by Aharoni, Ron, Berger, Eli, Kotlar, Dani +1 · 1 citation
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1605.05667
We study conjectures relating degree conditions in 3-partite hypergraphs to the matching number of the hypergraph, and use topological methods to prove special cases. In particular, we prove a strong version of a theorem of Drisko \citedrisko (as generalized by the first two authors \citeab), that every family of 2n-1 matchings of size n in a bipartite graph has a partial rainbow matching of size n. We show that milder restrictions on the sizes of the matchings suffice. Another result that is strengthened is a theorem of Cameron and Wanless \citeCamWan, that every Latin square has a diagonal (permutation submatrix) in which no symbol appears more than twice. We show that the same is true under the weaker condition that the square is row-Latin.