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 $k>0$ (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.