I readed in the book of Schrijver on matching matroid problem. Follow the figure:
I understood that this problem is a generalization of graph matching problem applied in the matroid $M = (S, 2^{S})$ and any graph. But I did not understand the part on the matroid intersection problem! Someone can explain?
The matroid intersection problem is to find a maximum (in size) independent set in two matroids $M_1$ and $M_2$, both in the same ground set. That is find $$\text{argmax} _{X\in M_1\cap M_2}|X|.$$
A regular of degree one graph, that is $deg(x)=1$ for all $x\in V_G$, is bipartite: it is a forest and forests are bipartite cause there are no cycles so no problem at all.
Call $U$ the vertices that are colored with the first color and $V$ the ones on the second color. Consider now the matroids $M_1$ which has as independents sets edges that do not share the same point in $U$ and, analogously, the independents in $M_2$ are edges that do not share a point in $V$. Then if an independent set is in $M_1$ and in $M_2$, then this set is a matching and viceversa.
Edit: Let $M_1,M_2$ be Matroids in the same ground set $S$ and consider a copy of $S$ given by $S'=\{a':a\in S\}$. Consider the bipartite graph $G=(S\cup S',E=\{\{a,a'\}:a\in S\})$. This is a regular bipartite graph. Consider the matroid given by $$M = (S\cup S',\{I\cup J: I\in M_1,J\in M_2\}),$$ in the notes we want to find $V\subseteq E$ such that (notice that this is a matching already for every possible $V\subseteq E$) such that $X=\bigcup _{v\in V} v\in M$ but notice that $X$ is of the form $X=Y\cup Y'$ with $Y'=\{a':a\in Y\}$ and $Y\in M_1$ and $Y'\in M_2$, but this means that $Y$ is independent in both $M_1$ and $M_2$ so we are solving the intersection problem.
