Proving K-Extendability of a Bipartite Graph

[Aryth1]

I was asked to see if I could prove this was true, but I have been totally unable to. I can't find many results about extendability and so I have had a lot of trouble. After days of thinking about it without anything to show, I figured that I'd ask for some help here. Here's the problem:

A graph is k-extendable if $|V(G)|\geq 2k + 2$ and for any k disjoint edges, there is a perfect matching containing them. Let $G[A,B]$ be a bipartite graph with a perfect matching. If for any $S\subset A$, $|N(S)|\geq |S| + k$, show that $G$ is k-extendable.

Any help is greatly appreciated!

 

[Fallen Angel]

Hi Aryth,

I think I'm missing something, if there is a perfect matching in a bipartite graph then $$|A|=|B|$$ and $$N(A)=B$$.

And from $$|N(S)| \geq |S| + k$$ with $$S=A$$ we get $$k\leq 0$$ so the statement is trivial.

What's wrong? :/