Degree conditions sufficient for Hall's theorem Let $G=(L,R,E)$ be a bipartite graph, are there conditions on the degree of the vertices under which the condition of Hall's theorem is surely satisfied? (meaning a perfect matching exists in the graph).

Konig's theorem proves that every $k$-regular bipratite graph has exactly $k$ edge disjoint perfect matchings, so the answer is obviously yes for any $k$-regular bipartite graph with $k&gt;0$ (it's not hard to see why hall's condition is satisfied in this case).

Is there a wider rule? 

For example: Let $|L|=|R|=n$, What is the minimal degree as a function of $n$ of every vertex in the graph such that the graph must admit a perfect matching?

I'm also wondering if these conditions will hold in general graphs (satisfying Tutte's theorem), and if not, then are there similar conditions for general graphs.
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 $S$.