For a finite graph G, we study the maximum 2-edge colorable subgraph problem and a related ratio $\frac{\mathrm{\mu }(G)}{\mathrm{\nu }(G)}$, where $\mathrm{\nu }(G)$ is the matching number of G, and $\mathrm{\mu }(G)$ is the size of the largest matching in any pair $(H,H^{\prime })$ of disjoint matchings maximizing $|H|+|H^{\prime }|$ (equivalently, forming a maximum 2-edge colorable subgraph). Previously, it was shown that $\frac{4}{5}\le \frac{\mathrm{\mu }(G)}{\mathrm{\nu }(G)}\le 1$, and the class of graphs achieving $\frac{4}{5}$ was completely characterized. In this paper, we first show that graph decompositions into paths and even cycles provide a new way to study these parameters. We then use this technique to characterize the graphs achieving $\frac{\mathrm{\mu }(G)}{\mathrm{\nu }(G)}=1$ among all graphs that can be covered by a certain choice of a maximum matching and H, $H^{\prime }$ as above.