  Given a graph G and a coloring of its edges, a subgraph of G is called
rainbow if its edges have distinct colors. The rainbow girth of an edge
coloring of G is the minimum length of a rainbow cycle in G. A generalization
of the famous Caccetta-Haggkvist conjecture (CHC), proposed by the first
author, is that if G has n vertices, G is n-edge-colored and the size of every
color class is k, then the rainbow girth is at most \lceil \frac{n}{k} \rceil.
In the only known example showing sharpness of this conjecture, that stems from
an example for the sharpness of CHC, the color classes are stars. This suggests
that in the antipodal case to stars, namely matchings, the result can be
improved. Indeed, we show that the rainbow girth of n matchings of size at
least 2 is O(\log n), as compared with the general bound of \lceil \frac{n}{2}
\rceil.
