Degree conditions sufficient for Hall's theorem
Are you by any chance looking for Hall's theorem ? It gives a complete characterization of when a bipartite graph admits a perfect matching. Specifically:

G admits a perfect matching if and only if for all subsets $S \subset L$, we have 
$$ |\Gamma(S)| \ge |S|$$ where $\Gamma(S) \subset R$ is the set of neighbors of vertices in $S$.