Conditions on Bipartite Graphs guaranteeing matchings

Question

Let $G=(X,Y)$ be a bipartite graph. I would like to collect a list of conditions that one can place on $X,Y$ (or $G$) that make Hall's condition true (that is, that for all $A\subset X$, the neighbourhood $\Gamma(A)$ is at least as large as $A$), so that a matching from $X$ to $Y$ exists.

A few examples I have are:

1. The bipartite graph is regular (and contains an edge).
2. $0$ is not an eigenvalue of $G$.
3. Every $x\in X$ has degree at least $1$, and also that whenever $xy\in E(G), d(x)\geq d(y)$.

Which other conditions do people know which I could add to this list?

Answer 1

4. If $|X|=|Y|=n$ and $\delta \geq n/2$.

5. If $|X|=|Y|=n$ and $\varepsilon \geq n^2-n+1$.

6. If $|X|=|Y|=n \ge 1$ Suppose that $X=\{x_1,x_2,..,x_n\}$ and $Y=\{y_1,y_2,...,y_n\}$. Two vertices $x_i$ and $y_j$ are adjacent in $G$ if and only if $i+j \ge n+1$.

Answer 2

Here are a few conditions that are equivalent to having an $X$-saturating matching. (This is stronger than what the question asks for, but is still an answer.)

1. $G$ has no vertex cover smaller than $X$.

2. $G$ has no independent set bigger than $Y$. (This follows from 1 because the complement of a vertex cover is an independent set.)

3. For all $A \subseteq Y$, $|\Gamma(A)| \ge |A|+|X|-|Y|$. (A Hall-like condition for $Y$ rather than $X$.)