  For a finite graph $G$, we study the maximum $2$-edge colorable subgraph
problem and a related ratio $\frac{\mu(G)}{\nu(G)}$, where $\nu(G)$ is the
matching number of $G$, and $\mu(G)$ is the size of the largest matching in any
pair $(H,H')$ of disjoint matchings maximizing $|H| + |H'|$ (equivalently,
forming a maximum $2$-edge colorable subgraph). Previously, it was shown that
$\frac{4}{5} \le \frac{\mu(G)}{\nu(G)} \le 1$, and the class of graphs
achieving $\frac{4}{5}$ was completely characterized. We show here that any
rational number between $\frac{4}{5}$ and $1$ can be achieved by a connected
graph. Furthermore, we prove that every graph with ratio less than $1$ must
admit special subgraphs.
