Tutte's Theorem: For every subset $S \subseteq V$, the number of odd components of $G-S$ is at most $|S|$.

Dirac's Theorem: Minimum degree $\delta(G) \geq n/2$.

Hall's Marriage Theorem: For every subset $X$ of the left partition, $|N(X)| \geq |X|$.

The graph must be 3-regular and bipartite.