Large rainbow matchings in general graphs

By a theorem of Drisko, any $2n-1$ matchings of size $n$ in a bipartite graph have a partial rainbow matching of size $n$. Inspired by discussion of Bar\'at, Gy\'arf\'as and S\'ark\"ozy, we conjecture that if $n$ is odd then the same is true also in general graphs, and that if $n$ is even then $2n$ matchings of size $n$ suffice. We prove that any $3n-2$ matchings of size $n$ have a partial rainbow matching of size $n$.

[1]  Ron Aharoni,et al.  Uniqueness of the extreme cases in theorems of Drisko and Erdős-Ginzburg-Ziv , 2018, Eur. J. Comb..

[2]  Gábor N. Sárközy,et al.  Rainbow matchings in bipartite multigraphs , 2015, Period. Math. Hung..

[3]  Ron Aharoni,et al.  Degree Conditions for Matchability in 3-Partite Hypergraphs , 2018, J. Graph Theory.

[4]  Arthur A. Drisko Transversals in Row-Latin Rectangles , 1998, J. Comb. Theory, Ser. A.

[5]  Ron Aharoni,et al.  Rainbow Matchings in r-Partite r-Graphs , 2009, Electron. J. Comb..

[6]  Gil Kalai,et al.  A topological colorful Helly theorem , 2005 .

[7]  Jens Vygen,et al.  The Book Review Column1 , 2020, SIGACT News.

[8]  J. Edmonds Paths, Trees, and Flowers , 1965, Canadian Journal of Mathematics.