Chapter

Theoretical Aspects of Computer Science

Volume 2292 of the series Lecture Notes in Computer Science pp 84-112

Date:

The Regularity Lemma and Its Applications in Graph Theory

  • János KomlósAffiliated withRutgers UniversityHungarian Academy of Sciences
  • , Ali ShokoufandehAffiliated withDepartment of Mathematics and Computer Science, Drexel University
  • , Miklós SimonovitsAffiliated withRutgers UniversityHungarian Academy of Sciences
  • , Endre SzemerédiAffiliated withRutgers UniversityHungarian Academy of Sciences

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Abstract

Szemerédi’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. Komlós 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.