Let $G$ be a bipartite graph with vertex classes $X$ and $Y$. What is a matching from $X$ to $Y$ ?

Show that if $|\Gamma(A)| \geqslant|A|$ for all $A \subset X$ then $G$ contains a matching from $X$ to $Y$.

Let $d$ be a positive integer. Show that if $|\Gamma(A)| \geqslant|A|-d$ for all $A \subset X$ then $G$ contains a set of $|X|-d$ independent edges.

Show that if 0 is not an eigenvalue of $G$ then $G$ contains a matching from $X$ to $Y$.

Suppose now that $|X|=|Y| \geqslant 1$ and that $G$ does contain a matching from $X$ to $Y$. Must it be the case that 0 is not an eigenvalue of $G$ ? Justify your answer.