Let $G=(X,Y,E)$ be an $r$-regular bipartite graph where $|X|=|Y|$ . Let $\phi$ an automorphism of $G$ with $\phi(X) = Y$ and $\phi \circ \phi = id$, and let $\psi$ be the mapping induced by $\phi$ on the edges of $G$.

Does $G$ have a matching $M$ such that $\psi(M) \cap M = \emptyset$ ? If so, how many of them it takes to cover the edges of $G$?