Semi-regular bipartite graphs and an extension of the marriage lemma

The well-known Marriage Lemma states that a bipartite regular graph has a perfect matching. We define a bipartite graph G with bipartition (X, Y) to be semi-regular if both x --> deg x, x is an element of X and y --> deg y, y is an element of Y are constant. The purpose of this note is to show that if C is bipartite and semi-regular, and if \X\ < \Y\, then there is a matching which saturates \X\. (Actually, we prove this for a condition weaker than semi-regular.) As an application, we show that various subgraphs of a hypercube have saturating matchings. We also exhibit classes of bipartite graphs, some of them semi-regular, whose vertices are the vertices of various weights in the hypercube Q(n) but which are not subgraphs of Q(n).