Prove that a bipartite graph $G = \left(V,E\right)$ has a perfect matching $\iff$ $\vert N(S)\vert\geq \vert S \vert $ for all $S \subseteq V$. (For any set $S$ of vertices in $G$ we define the neighbor set $N(S)$ of $S$ in $G$ to be the set of all vertices adjacent to vertices in $S$.) Also give an example to show that the above statement is invalid if the condition that the graph be bipartite is dropped.
[Math] Perfect matching in bipartite graphs
bipartite-graphsgraph theorymatching-theory
Best Answer
Let $X\uplus Y = V$ be the bipartition of $G$.
Hint:
I hope it helps $\ddot\smile$