As proved in [2], every epsilon-regular graph G on vortex set V-1 boolean O
R V-2, \V-1\ = \V-2\ = n, With density d > 2 epsilon and all vertex degrees
not too far from d, has about as many perfect matchings as a corresponding
random bipartite graph, i.e. about d(n)n!
In this paper we utilize that result to prove that with probability quickly
approaching one, a perfect matching drawn randomly from G is spread evenly
, in the sense that for any large subsets of vertices S subset of V-1 and T
C V-2, the number of edges of the matching spanned between S and T is close
to \S\\T\/n (c.f. Lemma 1).
As an application we give an alternative proof of the Blow-up Lemma of Koml
os, Sarkozy and Szemeredi [10].