Maximum Rainbow Matching is polynomial-time solvable if almost every color class is complete multipartite and NP-hard otherwise.
Monochromatic and heterochromatic subgraphs in edge-colored graphs - A survey.Graphs Comb., 24(4):237–263, 2008
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A Complexity Dichotomy for Generalized Rainbow Matchings Based on Color Classes
Maximum Rainbow Matching is polynomial-time solvable if almost every color class is complete multipartite and NP-hard otherwise.