Abstract
Drisko proved that 2n−1 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 2n−1 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 3n−2 to 3n−3, using a new line of proof that involves analysis of the appearance of badges. We also prove a “cooperative” generalization: for t>0 and n≥3, any 3n−4+t sets of edges, the union of every t of which contains a matching of size n, have a rainbow matching of size n.
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