I'm studying for a test in Graph Theory and I don't know how to prove that Union and intersection part of the following exercise:
Let $M$ and $N$ be matchings in a graph $G$ with $|M|>|N|$. Then there exist matchings $M'$ and $N'$ in $G$ such that $|M'|=|M|-1, |N'|=|N|+1, M' \cap N' = M \cap N$ and $M' \cup N'=M \cup N$.
What I have until now: I know that as $|M| > |N|$ then $N$ is not a maximum matching, so there is an $N$-augmenting path and therefore there is an $N'$ such that $|N'| = |N|+1$. I think I can see $N= M'$ and $M=N'$ but I don't know how to prove it formally.
Let call $T$ to $G[M\Delta N]=M\cup N−M\cup N$ Which each vertex in $T$ has a degree either one or two. Also, any component of $T$ is either an even cycle or a path with edges alternately in $M$ and $N$, where at least one path have starting and ending in $M$ because of the condition $|M|> |N|$.
P.D. Some modifications of the question have been done according to the hints given by @LeenDroogendijk
HINT: consider the subgraph $H$ of $G$ induced by the edges of $M\cup N$. Note that the edges that belong to $M\cap N$ are "isolated edges" in $H$.
What can you say about the degree of the vertices of $H$?
What do the connected components of this graph look like?
Now use the fact that $|M|>|N|$.