Applications of the regularity lemma for uniform hypergraphs (original) (raw)

Regularity Lemma for k-uniform hypergraphs

Random Structures and Algorithms, 2004

Szemerédi's Regularity Lemma proved to be a very powerful tool in extremal graph theory with a large number of applications. Chung [Regularity lemmas for hypergraphs and quasi-randomness, Random Structures and Algorithms 2 (1991), 241-252], Frankl and Rödl [The uniformity lemma for hypergraphs, Graphs and Combinatorics 8 (1992), 309-312, Extremal problems on set systems, Random Structures and Algorithms 20 (2002), 131-164] considered several extensions of Szemerédi's Regularity Lemma to hypergraphs. In particular, [Extremal problems on set systems, Random Structures and Algorithms 20 , 131-164] contains a regularity lemma for 3uniform hypergraphs that was applied to a number of problems. In this paper, we present a generalization of this regularity lemma to k-uniform hypergraphs. Similar results were independently and alternatively obtained by W. T. Gowers.

An Erdős–Gallai type theorem for uniform hypergraphs

European Journal of Combinatorics

A well-known theorem of Erdős and Gallai [1] asserts that a graph with no path of length k contains at most 1 2 (k−1)n edges. Recently Győri, Katona and Lemons [2] gave an extension of this result to hypergraphs by determining the maximum number of hyperedges in an r-uniform hypergraph containing no Berge path of length k for all values of r and k except for k = r + 1. We settle the remaining case by proving that an r-uniform hypergraph with more than n edges must contain a Berge path of length r + 1. Given a hypergraph H, we denote the vertex and edge sets of H by V (H) and E(H) respectively. Moreover, let e(H) = |E(H)| and n(H) = |V (H)|. A Berge path of length k is a collection of k distinct hyperedges e 1 ,. .. , e k and k + 1 distinct vertices v 1 ,. .. , v k+1 such that for each 1 ≤ i ≤ k, we have v i , v i+1 ∈ e i. A Berge cycle of length k is a collection of k distinct hyperedges e 1 ,. .. , e k and k distinct vertices v 1 ,. .. , v k such that for each 1 ≤ i ≤ k − 1, we have v i , v i+1 ∈ e i and v k , v 1 ∈ e k. The vertices v i and edges e i in the preceding definitions are called the vertices and edges of their respective Berge path (cycle). The Berge path is said to start at the vertex v 1. We also say that the edges e 1 ,. .. , e k of the Berge path (cycle) span the set ∪ k i=1 e i. A hypergraph is called r-uniform, if all of its hyperedges have size r. Győri, Katona and Lemons determined the largest number of hyperedges possible in an r-uniform hypergraph without a Berge path of length k for both the range k > r + 1 and the range k ≤ r.

The Uniformity Lemma for hypergraphs

Graphs and Combinatorics, 1992

In 1973, E. Szemeredi proved a theorem which found numerous applications in extremal combinatorial problems--The Uniformity Lemma for Graphs. Here we consider an extension of Szemeredi's theorem to r-uniform hypergraphs.

On characterizing hypergraph regularity

Random Structures and Algorithms, 2002

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

C O ] 6 S ep 2 01 7 3-uniform hypergraphs and linear cycles

2018

Gyárfás, Győri and Simonovits [3] proved that if a 3-uniform hypergraph with n vertices has no linear cycles, then its independence number α ≥ 2n 5 . The hypergraph consisting of vertex disjoint copies of a complete hypergraph K 5 on five vertices shows that equality can hold. They asked whether this bound can be improved if we exclude K 5 as a subhypergraph and whether such a hypergraph is 2-colorable. In this paper, we answer these questions affirmatively. Namely, we prove that if a 3-uniform linear-cycle-free hypergraph doesn’t contain K 5 as a subhypergraph, then it is 2-colorable. This result clearly implies that its independence number α ≥ ⌈n 2 ⌉. We show that this bound is sharp. Gyárfás, Győri and Simonovits also proved that a linear-cycle-free 3-uniform hypergraph contains a vertex of strong degree at most 2. In this context, we show that a linear-cycle-free 3-uniform hypergraph has a vertex of degree at most n− 2 when n ≥ 10.

C-Perfect K-Uniform Hypergraphs

Ars Comb., 2006

In this paper we define the concept of clique number of uniform hypergraph and study its relationship with circular chromatic number and clique number. For every positive integer k,p and q, 2q ≤ p we construct a k-uniform hypergraph H with small clique number whose circular chromatic number equals p q . We define the concept and study the properties of c-perfect k-uniform hypergraphs .

Intersection Graphs of k-uniform Linear Hypergraphs

European Journal of Combinatorics, 1982

A finite hypergraph H is said to be linear if every pair of distinct vertices of H is in at most one edge of H. A 2-uniform linear hypergraph is called a graph. The edge-degree of an edge of a graph G is the number of triangles in G containing the given edge. In this paper it is proved that there is a finite family IF of graphs such that any graph G with minimum degree at least 69 is the intersection graph of a 3-uniform linear hypergraph if and only if G has no induced subgraph isomorphic to a member of IF. Further, it is shown that there is a polynomial I(k) of degree less than or equal to 3 with the property that given any integer k (",,2) there exists a finite family IF(k) of graphs such that any graph G with minimum edge-degree at least I(k) is the intersection graph of a k-uniforrn linear hypergraph if and only if G has no induced subgraph isomorphic to a member of IF(k).

From The Cover: The hypergraph regularity method and its applications

Proceedings of the National Academy of Sciences, 2005

Szemeré di's regularity lemma asserts that every graph can be decomposed into relatively few random-like subgraphs. This random-like behavior enables one to find and enumerate subgraphs of a given isomorphism type, yielding the so-called counting lemma for graphs. The combined application of these two lemmas is known as the regularity method for graphs and has proved useful in graph theory, combinatorial geometry, combinatorial number theory, and theoretical computer science. Here, we report on recent advances in the regularity method for k-uniform hypergraphs, for arbitrary k > 2. This method, purely combinatorial in nature, gives alternative proofs of density theorems originally due to E. Szemeré di, H. Furstenberg, and Y. Katznelson. Further results in extremal combinatorics also have been obtained with this approach. The two main components of the regularity method for k-uniform hypergraphs, the regularity lemma and the counting lemma, have been obtained recently: Rö dl and Skokan (based on earlier work of Frankl and Rö dl) generalized Szemeré di's regularity lemma to k-uniform hypergraphs, and Nagle, Rö dl, and Schacht succeeded in proving a counting lemma accompanying the Rö dl-Skokan hypergraph regularity lemma. The counting lemma is proved by reducing the counting problem to a simpler one previously investigated by Kohayakawa, Rö dl, and Skokan. Similar results were obtained independently by W. T. Gowers, following a different approach.

The de Bruijn-Erdos Theorem for hypergraphs

2010

We also give an absolute lower bound cp(n,r)geqnchooser/q+r−1chooser\cp(n,r) \geq {n \choose r}/{q + r - 1 \choose r}cp(n,r)geqnchooser/q+r−1chooser when n=q2+q+r−1n = q^2 + q + r - 1n=q2+q+r−1, and for each rrr characterize the finitely many configurations achieving equality with the lower bound. Finally we note the connection of cp(n,r)\cp(n,r)cp(n,r) to extremal graph theory, and determine some new asymptotically sharp bounds for the Zarankiewicz problem.

Hypergraph Extensions of the Erdős-Gallai Theorem

Electronic Notes in Discrete Mathematics, 2010

We extend the Erdős-Gallai Theorem for Berge paths in r-uniform hypergraphs. We also find the extremal hypergraphs avoiding t-tight paths of a given length and consider this extremal problem for other definitions of paths in hypergraphs.