Regularity Lemma For Graphs Research Articles

An Algorithmic Version of the Hypergraph Regularity Method

Extending the Szemerédi regularity lemma for graphs, P. Frankl and V. Rödl [Random Structures Algorithms, 20 (2002), pp. 131–164] established a 3-graph regularity lemma triple systems ${\cal G}_n$ admit bounded partitions of their edge sets, most classes of which consist of regularly distributed triples. Many applications of this lemma require a companion counting lemma [B. Nagle and V. Rödl, Random Structures Algorithms, 23 (2003), pp. 264–332] allowing one to find and enumerate subhypergraphs of a given isomorphism type in a “dense and regular” environment created by the 3-graph regularity lemma. Combined applications of these lemmas are known as the 3-graph regularity method. In this paper, we provide an algorithmic version of the 3-graph regularity lemma which, as we show, is compatible with a counting lemma. We also discuss some applications.

Regular Partitions of Hypergraphs: Regularity Lemmas

Szemerédi's regularity lemma for graphs has proved to be a powerful tool with many subsequent applications. The objective of this paper is to extend the techniques developed by Nagle, Skokan, and the authors and obtain a stronger and more ‘user-friendly’ regularity lemma for hypergraphs.

Counting Small Cliques in 3-uniform Hypergraphs

Many applications of Szemerédi's Regularity Lemma for graphs are based on the following counting result. If , there is a possibility that the approach of this paper can be adopted to the general case as well.