We first give two different formulations of a theorem known as P. Hall's marriage theorem. We give a constructive proof and an enumerative one. If A is a subset of the vertices of a graph, then denote by г(A) the set ∪a∈Aг(a). Consider a bipartite graph G with vertex set X ∪ Y (every edge has one endpoint in X and one in Y). A matching in G is a subset E1 of the edge set such that no vertex is incident with more than one edge in E1. A complete matching from X to Y is a matching such that every vertex in X is incident with an edge in E1. If the vertices of X and Y are thought of as boys and girls, respectively, or vice versa, and an edge is present when the persons corresponding to its ends have amicable feelings towards one another, then a complete matching represents a possible assignment of marriage partners to the persons in X.

Theorem 5.1.A necessary and sufficient condition for there to be a complete matching from X to Y in G is that |г(A) > |A| for every A ⊆ X.

Proof: (i) It is obvious that the condition is necessary.

(ii) Assume that |г(A)| ≥ |A| for every A ⊆ X. Let |X| = n, m < n, and suppose we have a matching M with m edges. We shall show that a larger matching exists.