This is a question about the proof of the converse of Theorem 5.2 in "Graph Theory with Applications" by J.A. Bondy and U.S.R. Murty, 1976, North-Holland.
Theorem. Let $G$ be a bipartite graph with bipartition $(X,Y)$. Then $G$ contains a matching that saturates every vertex in $X$ if and only if
\begin{align*} \forall S \subset X : \left|N(S)\right| \geq \left|S\right| \tag{5.2} \end{align*} Proof. [...]
Conversely, suppose that $G$ is a bipartite graph satisfying (5.2), but that $G$ contains no matching saturating all the vertices in $X$. We shall obtain a contradiction. Let $M*$ be a maximum matching in $G$. By our supposition, $M*$ does not saturate all vertices in $X$. Let $u$ be an $M*$-unsaturated vertex in $X$, and let $Z$ denote the set of all vertices connected to $u$ by $M*$-alternating paths. Since $M*$ is a maximum matching, it follows from theorem 5.1 that $u$ is the only $M*$-unsaturated vertex in $Z$. [...]
Question. How come $u$ is in $Z$? $Z$ is the set of all vertices connected to $u$. But $u$ is not connected to itself because $G$ is bipartite. So there's no way $u$ could be in $Z$. What's going on there?
Thank you!
