On characterizing hypergraph regularity

Abstract

Szemerédi'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 ⑀-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 ⑀-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 293conditions 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].

Loose Hamilton cycles in 3-uniform hypergraphs of high minimum degree

“…Such an algorithmic version is known for the case r = 1 of Lemma 5.8 (see [8]). As mentioned after Lemma 5.19, the only place where we use the (δ * , r)-regularity instead of just the (δ * , 1)-regularity is Lemma 5.19 due to Dementieva et al [4]. In fact, in [4] the authors conjecture that Lemma 5.19 even holds for r = 1, i.e.…”

“…As mentioned after Lemma 5.19, the only place where we use the (δ * , r)-regularity instead of just the (δ * , 1)-regularity is Lemma 5.19 due to Dementieva et al [4]. In fact, in [4] the authors conjecture that Lemma 5.19 even holds for r = 1, i.e. if we only assume that the triad P is (δ * , 1)-regular.…”

“…Part (i) of Lemma 5.19 was proved by Frankl and Rödl [5] (see also Nagle and Rödl [19]), part (ii) is due to Dementieva et al [4].…”

“…However, it seems that such an extension of our results will not be quite straightforward. For example, we use some results of Dementieva et al [4] for which analogues for r-uniform hypergraphs do not exist yet.…”

Counting in hypergraphs via regularity inheritance

“…This appeal of working with the weaker 3-graph regularity is a local characterisation in terms of subgraph counts. This characterisation does not hold for large values of r in the stronger 3-graph regularity, thus working with r = 1 recovers the useful property; see [4,10].…”