Let $k$ be a natural number with $k \geq 2$. Consider a bipartite graph $G$ with vertex classes $A$ and $B$, with $|B| = k|A|$. Use Hall's Theorem applied to a suitable graph $G'$ to show that if $|N(S)| \geq k|S|$ for all sets of vertices $S$ in $A$ with $S$ not the empty set, then the vertices of $G$ can be covered entirely by disjoint $k$-stars.
I have no idea where to even begin.
Thanks :)
Edit: I now know that I need to make a graph $G'$, with vertex classes $A'$ and $B$, where $A'$ is obtained by multiplying the vertices of $A$ by $k$. But then I am not sure where to go from there? I know the aim is to say that Hall's Marriage Theorem holds for $G'$, so when we 'collapse' $A'$ down to $A$, we get what we want ($G$ covered entirely by disjoint $k$-stars), but I am not sure how to show that this is the case?