A matching M in an edge–colored (hyper)graph is rainbow if each pair of edges in M have distinct colors. We extend the result of Erdos and Spencer on the existence of rainbow perfect matchings in the complete bipartite graph Kn,n to complete bipartite multigraphs, dense regular bipartite graphs and complete r-partite r-uniform hypergraphs. The proof of the results use the Lopsided version of the Local Lovász Lemma.
Cano, M., Perarnau, G., Serra, O. Rainbow perfect matchings in r-partite graph structures. "Electronic notes in discrete mathematics", 3 Octubre 2016, vol. 54, p. 193-198.