Past exam of the graph Theory and Probabilistic Combinatorics course of the University of Oxford / 2011 / A / 1 / b / iii

Hall's marriage theorem says that a bipartite graph with parts $L,R$ has a matching saturating $L$ if and only ifNecessity follows because the matched neighbours of vertices of $S$ are distinct and belong to $N(S)$.

For sufficiency, assume these inequalities. Let $K=U\cup W$ be any vertex cover, with $U\subseteq L$ and $W\subseteq R$. Every neighbour of $S=L\setminus U$ must belong to $W$, since otherwise its edge from $S$ is uncovered. Therefore $|W|\ge |N(S)|\ge |S|$, and $|K|\ge |U|+|S|=|L|$. The whole part $L$ is itself a vertex cover of size $|L|$. Part (ii) now gives a matching of size exactly $|L|$, so it saturates $L$.