 Solution

Take a disjoint union of $n$ single edges, each joining one vertex of $X$ to one of $Y$. Its unique matching in a graph covering $X$ contains every edge, while any singleton $A\subset X$ satisfies $|\Gamma (A)|=|A|=1$. For $n\ge 2$ this $A$ is nonempty and proper. This illustrates why connectedness was needed in part (ii).