Rainbow Paths and Large Rainbow Matchings

A conjecture of the first two authors is that $n$ matchings size in any graph have a rainbow matching $n-1$. We prove lower bound $\frac{2}{3}n-1$, improving on trivial $\frac{1}{2}n$, and an analogous result for hypergraphs. For $\{C_3,C_5\}$-free graphs disjoint we obtain $\frac{3n}{4}-O(1)$. also discuss alternating paths, if true would yield $n-\sqrt{2n}$. non-alternating (ordinary paths) version this conjecture.