The Regularity Lemma and Its Applications in Graph Theory

Authors:
János Komlós;Ali Shokoufandeh;Miklós Simonovits;Endre Szemerédi
Affiliations:
-;-;-;-
Venue:
Theoretical Aspects of Computer Science, Advanced Lectures [First Summer School on Theoretical Aspects of Computer Science, Tehran, Iran, July 2000]
Year:
2000

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Abstract

Szemer茅di's Regularity Lemma is an important tool in discrete mathematics. It says that, in some sense, all graphs can be approximated by random-looking graphs. Therefore the lemma helps in proving theorems for arbitrary graphs whenever the corresponding result is easy for random graphs. In the last few years more and more new results were obtained by using the Regularity Lemma, and also some new variants and generalizations appeared. Koml贸s and Simonovits have written a survey on the topic [96]. The present survey is, in a sense, a continuation of the earlier survey. Here we describe some sample applications and generalizations. To keep the paper self-contained we decided to repeat (sometimes in a shortened form) parts of the first survey, but the emphasis is on new results.