# Generalizing Hall's marriage theorem

Fix positive integers $$m,n,k$$ such that $$n\geq k$$.

Consider a bipartite graph between two sets of vertices $$A$$ and $$B$$ consisting of $$n$$ and $$mn$$ vertices respectively. Suppose each vertex in $$A$$ is connected to exactly $$mk$$ vertices in $$B$$ and each vertex in $$B$$ is connected to exactly $$k$$ vertices in $$A$$. (Assume there are no double edges between any pair of vertices.)

Is it possible to select a subset $$B'$$ of $$B$$ of size $$n$$ such that each vertex of $$A$$ is connected to exactly $$k$$ vertices in $$B'$$?

This is trivially true when $$k=1$$, whereas the case $$k=2$$ is equivalent to Hall's marriage theorem. (This follows from Peterson's 2-factor theorem.) Can we say something for general $$k$$?