Regularity Lemma For Hypergraphs Research Articles

On characterizing hypergraph regularity

AbstractSzemerédi's Regularity Lemma is a well‐known and powerful tool in modern graph theory. This result led to a number of interesting applications, particularly in extremal graph theory. A regularity lemma for 3‐uniform hypergraphs developed by Frankl and Rödl [8] allows some of the Szemerédi Regularity Lemma graph applications to be extended to hypergraphs. An important development regarding Szemerédi's Lemma showed the equivalence between the property of ϵ‐regularity of a bipartite graph G and an easily verifiable property concerning the neighborhoods of its vertices (Alon et al. [1]; cf. [6]). This characterization of ϵ‐regularity led to an algorithmic version of Szemerédi's lemma [1]. Similar problems were also considered for hypergraphs. In [2], [9], [13], and [18], various descriptions of quasi‐randomness of k‐uniform hypergraphs were given. As in [1], the goal of this paper is to find easily verifiable conditions for the hypergraph regularity provided by [8]. The hypergraph regularity of [8] renders quasi‐random “blocks of hyperedges” which are very sparse. This situation leads to technical difficulties in its application. Moreover, as we show in this paper, some easily verifiable conditions analogous to those considered in [2] and [18] fail to be true in the setting of [8]. However, we are able to find some necessary and sufficient conditions for this hypergraph regularity. These conditions enable us to design an algorithmic version of a hypergraph regularity lemma in [8]. This algorithmic version is presented by the authors in [5]. © 2002 Wiley Periodicals, Inc. Random Struct. Alg., 21: 293–335, 2002

An Algorithmic Regularity Lemma for Hypergraphs

In this paper, we will consider the problem of designing an efficient algorithm that finds an $\epsilon$-regular partition of an l-uniform hypergraph.

Ramsey Properties of Random Hypergraphs

LetK(k)(n,p) be the randomk-uniform hypergraph obtained by independent inclusion of each of the[formula]k-tuples with probabilityp. For an arbitraryk-uniform hypergraphGand every integerrwe find the threshold for the property that everyr-coloring of the vertices ofK(k)(n,p) results in a monochromatic copy ofG. In the edge coloring case, which is of our main interest, we find the threshold only fork=3,r=2, andG=K34. In the proof we utilize a recent version of the hypergraph regularity lemma due to Frankl and Rödl.

Regularity lemmas for hypergraphs and quasi‐randomness

AbstractWe give a simple proof for Szemerédi's Regularity Lemma and its generalization for k‐uniform hypergraphs. For fixed k, there are altogether k ‐1 different versions of the regularity lemma for k‐uniform hypergraphs. The connection between regularity lemmas for hypergraphs and quasi‐random classes of hypergraphs is also investigated.