Perfect Matchings in ε-Regular Graphs and the Blow-Up Lemma


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 [10].

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Received: December 5, 1997

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Rödl, V., Ruciński, A. Perfect Matchings in ε-Regular Graphs and the Blow-Up Lemma. Combinatorica 19, 437–452 (1999).

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  • AMS Subject Classification (1991) Classes:  05C75