Drisko proved that 2nβ1 matchings of size n in a bipartite graph have a
rainbow matching of size n. For general graphs it is conjectured that 2n
matchings suffice for this purpose (and that 2nβ1 matchings suffice when n
is even). The known graphs showing sharpness of this conjecture for n even
are called badges. We improve the previously best known bound from 3nβ2 to
3nβ3, using a new line of proof that involves analysis of the appearance of
badges. We also prove a "cooperative" generalization: for t>0 and nβ₯3,
any 3nβ4+t sets of edges, the union of every t of which contains a matching
of size n, have a rainbow matching of size n.Comment: Accepted for publication in Discrete Mathematics. 19 pages, 2 figure