Matchings Meeting Quotas and their Impact on the Blow-Up Lemma

A bipartite graph G = (U, V ; E) is called #-regular if the edge density of every su#ciently large induced subgraph di#ers from the edge density of G by no more than #. If, in addition, the degree of each vertex in G is between (d #)n and (d + #)n, where d is the edge density of G and = n, then G is called super (d, #)-regular. In [Combinatorica, 19 (1999), pp. 437-- 452] it was shown that if S U and T V are subsets of vertices in a super-regular bipartite graph G = (U, V ; E), and if a perfect matching M of G is chosen randomly, then the number of edges of M that go between the sets S and T is roughly |S||T |/n. In this paper