Proof union of two disjoint perfect matchings in a bipartite graph is covering all vertices of G with disjoint cycles

Question

I am trying to prove that the union of two disjoint perfect matchings in a bipartite graph is covering all vertices of G with disjoint cycles.

I would appreciate if anyone knew the proof or had ideas about it

Answer 1

Let $(V_1,V_2)$ be the bipartition sets, $M_1$, $M_2$ the two matchings, and $G'$ the graph with vertex set $V_1\cup V_2$ and edge set $M_1\cup M_2$. Since $G$ is bipartite, each edge in a matching is incident on a vertex $u\in V_1$ and a vertex in $V_2$, so $|M_1|\leqslant|V_1|,|V_2|$. But $M_1$ is a perfect matching, so $|M_1|\geqslant|V_1|,|V_2|$, and so $|M_1|=|V_1|=|V_2|$ (and similarly $|M_2|=|M_1|$).

Since $M_1\cap M_2=\varnothing$, each vertex in $u\in G'$ is incident with an edge $uv\in M_1$ and some $uw\in M_2$. It follows that $G'$ is $2$-regular, and so is the union of disjoint cycles.