Badges and rainbow matchings

Drisko proved that 2n−1 matchings of size n in a bipartite graph have rainbow matching n. For general graphs it is conjectured 2n suffice for this purpose (and when even). The known showing sharpness conjecture even are called badges. We improve the previously best bound from 3n−2 to 3n−3, using new line proof involves analysis appearance also prove “cooperative” generalization: t>0 and n≥3, any 3n−4+t sets edges, union every t which contains n,