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Given a graph $G$ which is bipartite and balanced and has unique perfect matching let $G^{e}$ be $G$ without edge $e$. Let $G\cup G_{\pi,\pi'}$ be union of $G$ and $G_{\pi,\pi'}$ where $G_{\pi,\pi'}$ ...

Given a bipartite graph we know that symmetric difference of any two perfect matchings is union of even cycles. Conversely when is it true that every union of even cycles comes from symmetric ...