Semi-Regular Bipartite Graphs and an Extension of the Marriage Lemma

Abstract

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 \(x \mapsto\) deg \(x,x \in X\) and \(y \mapsto\) deg \(y, y \in Y\) are constant. The purpose of this note is to show that if \(G\) 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 \(Q_n\).