Solution
= Solution
\b[No.] In the six-cycle with cyclic vertices $1,2,\ldots,6$, every edge belongs to one of the two alternating <perfect matchings>. However, the disjoint edges $12$ and $45$ cannot belong to a common <perfect matching>: the remaining vertices $3,6$ are not adjacent. Taking $X=\{1,3,5\}$ and $Y=\{2,4,6\}$ gives the required <bipartite graph> counterexample.