Perfect Matching in Bipartite Graph with mutually exclusive edges
It is $NP$ as $SAT$ can be polynomially reduced to this problem.

Let clauses be the left part vertexes, and literals be the right part vertexes. Draw an edge $(x,y)$ iff clause $x$ contains literal $y$. Finally, a pair of edges $(x_1, y_1)$ and $(x_2, y_2)$ will be exclusive iff $y_1$ corresponds to literal $a$ and $y_2$ corresponds to literal $\overline{a}$.