Topological methods for the existence of a rainbow matching

Abstract We present recent results regarding rainbow matchings in bipartite graphs. Using topological methods we address a known conjecture of Stein and show that if K n , n is partitioned into n sets of size n, then a partial rainbow matching of size 2n/3 exists. We generalize a result of Cameron and Wanless and show that for any n matchings of size n in a bipartite graph with 2n vertices there exists a full matching intersecting each matching at most twice. We show that any n matchings of size approximately 3n/2 have a rainbow matching of size n. Finally, we show the uniqueness of the extreme case for a theorem of Drisko and provide a generalization of Drisko's theorem.

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