I'm struggling on this question and any hints (not full answers) would be greatly appreciated.

Consider Bipartite Graph G with vertex Classes A and B of the same size. I need to use Hall's Theorem to show that is |N(S)| > |S| for all subsets S of A then for any edge e, G contains a perfect matching containing e.



Note that Hall's condition usually doesn't have a strict inequality. So try removing the specified edge and its endpoints from the graph. How much smaller could the neighborhood of any set $S\subseteq A$ have become?

  • $\begingroup$ Got it; so you can add an endpoint of e into S then either |N(s)| grows by 1 or stays the same, so Hall's condition holds since we had a strict inequality at first, and I can adapt the proof of Hall's theorem to show a perfect matching containing e exists, thanks! $\endgroup$ – Tim Hodgkin Oct 28 '15 at 14:31

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