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Consider a bipartite graph $G=(L\cup R,E)$ where $|L|=|R|=k$. The set of neighbors of a vertex $v\in V$ is defined as $N(v)$. The bipartite graph has the following properties:

1) $|N(v)|\geq 1$

2) $|\bigcup_{\forall l \in L}N(l)|=k$

Prove that a perfect matching exists for the bipartite graph $G$.

I know that I have to use Hall's marriage theorem here but I don't know how to prove that for all subsets of $L$, the number of neighbors is as big as the subsets for $G$.

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It is false for all $k\geq 3$. Consider the bipartite graph which has a special vertex $r\in R$ and $l\in L$ such that an edge is part of the graph if and only if $r$ or $l$ is one of the endpoints.

A counterexample for $k=3$:

enter image description here

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  • $\begingroup$ Could you please elaborate your answer with an example. $\endgroup$ Oct 18 '16 at 1:17

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