 Union of two matchings

In the spanning subgraph formed by the union of two matchings in a graph $M_1,M_2$, every vertex has vertex degree at most two. A common edge is an isolated two-vertex connected component of a graph. Every other nontrivial connected component of a graph is a graph path or an even graph cycle, with the edges alternating between the two matchings in a graph.

Suppose the original graph is a bipartite graph with vertex sets $A,B$, $M_1$ covers a specified subset $X\subseteq A$, and $M_2$ covers a specified subset $Y\subseteq B$. Their union contains a matching in a graph covering $X\cup Y$. On an even graph cycle, either alternating choice covers every vertex. On an odd-length graph path, the larger alternating choice covers every vertex. On an even-length graph path, both endpoints lie in the same side: choose $M_1$ if they lie in $A$, or $M_2$ if they lie in $B$. Every required endpoint is covered by that choice, and every internal vertex meets both choices. Select each common edge as well; isolated vertices are not in $X\cup Y$. Different connected components of a graph do not interfere.