# Hall's Marriage Theorem

I am aware that Hall's Marriage theorem for complete matching goes like "A bipartite graph $$G$$ with bipartition $$(V_1, V_2)$$ has a complete matching from $$V_1$$ to $$V_2$$ if and only if $$|N(A)| \geq |A|, \forall A \subseteq V_1$$

I want to know in which cases does an equality hold, i.e. $$|N(A)| = |A|, \forall A \subseteq V_1$$

Any help is greatly appreciated.

This can only hold if every vertex of $$V_1$$ has degree $$1$$ exactly (so the graph is a disjoint union of edges).
Why? Consider each vertex $$u\in V_1$$ as the singleton set $$A=\{u\}$$. If it has any number of neighbors not equal to one, you've already found a set breaking the equality.