Perfect Matchings in ε-Regular Graphs and the Blow-Up Lemma
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on vertex set \(\), \(\), with density d>2ε and all vertex degrees not too far from d, has about as many perfect matchings as a corresponding random bipartite graph, i.e. about \(\).
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 \(\) and \(\), 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 Komlós, Sárközy and Szemerédi .
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