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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.

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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.

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