Let $G=(A \cup B,E)$ be a bipartite graph with $|A| = |B| = n$. For a set $X \subseteq A$, let $N_i(X) \subseteq B$ be the set of vertices that have at least i neighbors in $X$.
I'd like to prove: If $|N_1(X)| + |N_2(X)| + ... + |N_k(X)| \geq k|X|$ for every set $X \subseteq A$ then $G$ has a $k$ - factor.
I'm aware, that it is enough to show this for connected graphs. The case $k=1$ is clear. For the case $k=2$ I tried to somehow use the idea of the proof of the Peterson Theorem but I failed to even construct an Eulerian tour. Can someone give me an idea how to handle this problem?