International Journal of Combinatorics

Volume 2014 (2014), Article ID 602657, 9 pages

http://dx.doi.org/10.1155/2014/602657

Research Article

A Weighted Regularity Lemma with Applications

Béla Csaba¹ and András Pluhár²

¹Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, Szeged 6724, Hungary

²Department of Computer Science, University of Szeged, Árpád tér 2, Szeged 6720, Hungary

Received 13 February 2014; Revised 27 May 2014; Accepted 27 May 2014; Published 19 June 2014

Academic Editor: Laszlo A. Szekely

Copyright © 2014 Béla Csaba and András Pluhár. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We prove an extension of the regularity lemma with vertex and edge weights which in principle can be applied for arbitrary graphs. The applications involve random graphs and a weighted version of the Erdős-Stone theorem. We also provide means to handle the otherwise uncontrolled exceptional set.

1. Introduction

Let be a bipartite graph. For let denote the number of edges with one endpoint in and the other in . Given an we say that the -pair is -regular if

for every , and , .

This definition plays a crucial role in the celebrated Regularity Lemma of Szemerédi; see [1, 2]. The regularity lemma is a very powerful tool when applied to a dense graph. It has found lots of applications in several areas of mathematics and computer science; for applications in graph theory see for example, [3]. However, it does not tell us anything useful when applied for a sparse graph (i.e., a graph on vertices having edges).

There has been significant interest to find widely applicable versions for sparse graphs. This turns out to be a very hard task. Kohayakawa [4] proved a sparse regularity lemma, and with Kohayakawa et al. [5] they applied it for finding arithmetic progressions of length 3 in dense subsets of a random set. In their sparse regularity lemma dense graphs are substituted by dense subgraphs of a random (or quasirandom) graph. Naturally, a new definition of -regularity was needed; below we formulate a slightly different version from theirs.

Let and be two bipartite graphs such that . We say that the -pair is -regular in relative to if

for every , and , . It is easy to see that the above is a generalization of -regularity; in the original definition the role of is played by the complete bipartite graph . In this more general definition can be a rather sparse graph; it only has to be dense relative to ; that is, should be a constant.

In this paper we further generalize the notion of quasirandomness and -regularity by introducing weighted regularity using vertex and edge weights. This enables us to prove a more general and perhaps more applicable regularity lemma. Let us remark that another notion of regularity is used by Alon et al. [6]; later we will discuss how their work relates to ours. A recent approach by Scott [7] defines regularity of matrices and deduces a regularity lemma for graphs via their adjacency matrices. This approach turns out to be less flexible than the one we choose in the present paper (for earlier versions, see [8]).

The basic tool is the Strong Structure Theorem of Tao [9], where he simplifies the proof of the original regularity lemma itself and gives new insights, too. Following his lines became technically feasible to extend regularity to the case when both the edges and the vertices of a graph are weighted (note that the measures are in close connection with each other.) We remark that similar ideas might be used to find a regularity lemma for sparse hypergraphs as well.

The structure of the paper is as follows. First we discuss weighted quasirandomness and weighted -regularity in the second section. In the third section we prove the new version of the regularity lemma. Finally, we show some applications in the fourth section; in particular, we prove a weighted version of the Erdős-Stone theorem.

2. Basic Definitions and Tools

Throughout the paper we apply the relation “”: if is sufficiently smaller than . This notation is widely applied in papers using the regularity lemma and simplifies our notation, too.

Let and be a graph on vertices. Set ; this is the density of . We define the density of the , pair of subsets of by . We say that is -quasi-random if it has the following property: If such that and then

That is, the edges of are distributed “randomly.” In order to formulate our regularity lemma we have to define quasirandomness in a more general way that admits weights on vertices and edges.

For a function and , is defined by the usual way; that is, . We will also use the indicator function of the edge set of a graph : and if  .

We define the weighted quasirandomness of a graph with given weight-functions and . For let

In particular, for . Observe that the function is an analogon of the vertex counting function on a set, while the function counts the edges in the unweighted case.

Definition 1. A graph is weighted -quasi-random with weight-function and if for any such that and , one has

Observe that choosing and gives back the first definition of quasirandomness. The notion of quasirandomness readily extends to bipartite (or multipartite) graphs. In that case the sets and are chosen from different classes. There is another weaker notion of quasirandomness, which we will also use.

Definition 2. Let be an absolute constant. A graph is weighted -quasi-random with weight-functions and if for any such that and , one has

Clearly, if a graph is -quasi-random and , then it is -quasi-random, where . Now we need to describe the weighted version of relative regularity.

Definition 3. Let and be graphs, , and assume that is a -quasi-random with weight functions and as defined above. For and the pair in is -weighted -regular relative to , or briefly weighted -regular, if

for every and provided that , . Here

Remarks. Note that weighted -regularity is nothing but the well-known -regularity when and and is chosen to be identically the reciprocal of the density of as before. Since we also have . Hence, the first inequality of the definition does not refer to explicitly but contains information on it.

Next we define weighted regular partitions.

Definition 4. Let and be graphs, and and weight functions. has a weighted -regular partition relative to if its vertex set can be partitioned into clusters such that

(i), (ii) for every , ,(iii)all but at most of the pairs for are weighted -regular in relative to .

In order to show our main result we will use the Strong Structure Theorem of Tao that allows a short exposition. In fact we will closely follow his proof for the regularity lemma as discussed in [9].

First we have to introduce some definitions. Let be a real finite-dimensional Hilbert space, and let be a set of basic functions or basic structured vectors of of norm at most 1. The function is -structured with the positive integers , if one has a decomposition

with and for . We say that is -pseudorandom for some if for all . Then we have the following.

Theorem 5 (Strong Structure Theorem—T. Tao). Let and be as above, let , and let be an arbitrary function. Let be such that . Then we can find an integer and a decomposition where (i) is -structured, (ii) is -pseudorandom, and (iii) .

Note that the proof of Theorem 5 yields a polynomial algorithm; hence, our regularity lemma has the same complexity.

3. Weighted Regularity Lemma Relative to a Quasirandom Graph

First we define the Hilbert space , and . We generalize Example 2.3 of [9] to weighted graphs. Let be a -quasi-random graph on vertices with weight functions and . Let be the -dimensional space of functions , endowed with the inner product

It is useful to normalize the vertex and edge weight functions; we assume that and . We also assume that for every . Observe that if then . We let be the collection of 0,1-valued functions for , , where if and only if and . We have the following.

Theorem 6 (weighted regularity lemma). Let and , such that , and let . If is a weighted -quasi-random graph on vertices with sufficiently large depending on and , , then admits a weighted -regular partition relative to into the partition sets such that for some constant .

Proof. Let us apply Theorem 5 to the function with parameters and function to be chosen later. We get the decomposition

where is -structured, is -pseudorandom, and with .

The function is the combination of at most basic functions:

where are subsets of and agrees with the indicator function of the edges of in between and . Any pair partitions into at most 4 subsets. Overall we get a partitioning of into at most subsets; we will refer to them as atoms. Divide every atom into subsets of total vertex weight , except possibly one smaller subset. The small subsets will be put into ; the others give , with . We refer to the sets for as clusters. If is sufficiently large then this partitioning is nontrivial. From the construction it follows that each is entirely contained within an atom. It is also clear that and for every .

We have that

From this and the normalization of it follows that

Clearly,

for all but at most pairs . If the above is satisfied for a pair then we call it a good pair. We will apply the Cauchy-Schwarz inequality. For that let and ; then

Since

we get that

if is a good pair.

Assume that is a good pair. From the pseudorandomness of we have that

for every and .

We will show that every good pair is weighted -regular in relative to . Let be a good pair, and assume that , and , . To show that is weighted -regular, it is sufficient to show that

Recall that

since .

Substituting for it is sufficient to verify the following inequalities:

For proving (22) recall that is constant on and -structured. Since the basic functions are -valued, we get that . Moreover, is -quasi-random, where . Therefore, (22) follows from the inequality , since .

The proof of (23) goes as follows. The first term is

and the second is

Noting that for we get that the sum of the above terms is at most

if .

For (24) first notice that it is upper bounded by

We also have that

by the normalization of and and from the fact that is quasirandom. From this it is easy to see that if then (24) is at most . This finishes the proof of the theorem.

4. Quasirandom Weightings and Applications

In this section we first prove that a random graph with widely differing edge probabilities is quasirandom, if none of the edge probabilities are too small. In this case the vertex weights will all be one, but edges will have different weights. Then we show examples where vertices have different weights. We will consider the relation of weighted regularity and volume regularity. We define the “natural weighting” of and prove a weighted version of the Erdős-Stone theorem for this weighting. Finally, we show how to partially control the nonexceptional set by natural weightings.

4.1. Quasirandomness in the Model

In this section we will prove that random graphs of the model are quasirandom in the strong sense with high probability. A special case of this model is the well-known model for random graphs. A regularity lemma for this case was first applied by Kohayakawa et al. [5]. They studied for in order to find arithmetic progressions of length three in dense subsets of random subsets of .

The model was first considered by Bollobás [10]. Recently it was also studied by Chung and Lu [11]. In this model one takes vertices and draws an edge between the vertices and with probability , randomly and independently of each other. Note that if , then we get back the well-known model. It is a straightforward application of the Chernoff bound that a random graph is quasirandom with high probability if . However, the case of is somewhat harder.

Lemma 7. Let . There exists a such that if and for every and , then is weighted -quasi-random with probability at least if is sufficiently large.

Proof. First of all let , and let . Set . Let , and let for . Let and be a pair of disjoint sets, both of size at least . We partition the pairs , where and , into disjoint sets : if then will belong to . Let . We will denote by .

We will prove that the following inequality holds with probability at least :

where is a random variable which is 1 if ; otherwise it is 0. This implies the quasirandomness of since there are less than pairs of disjoint subsets of . Observe that

Applying the large deviation inequalities and from [12], we are able to bound the number of edges in between and for the edges of in case is sufficiently large as follows. According to we have that

where

We estimate the exponent in case :

where we used the definition of . For being much less than , direct substitution gives a useless bound. For this case we have the useful inequality

where and . This implies that the exponent is at most even in case .

Indeed, let , , and be the events that , , and , respectively. Clearly and are independent, and . So we have ; that is, , since by

With this we have proved that the sum of the weights of the edges of will not be much larger than their expectation with high probability.

Now we estimate the probability that the sum of the weights is much less than their expectation. Let us use again directly to the sums over ’s:

The exponent in the inequality can be estimated very similarly as before:

moreover, this bound applies for an arbitrary .

Putting these together we have that

This implies that

where the last two inequalities hold with probability at least for a given pair of sets and if is sufficiently large. Since

the claimed bound follows with high probability.

Remark 8. It is very similar to prove that with high probability , we omit the details. From this it follows that rescaling the above edge weights by a factor of and letting provide us with -quasi-random weights for most graphs from such that and . That is, with high probability we can apply the regularity lemma for any , where .

4.2. Simple Examples for Defining Vertex and Edge Weights

When defining the notion of weighted quasirandomness and weighted regularity, we mentioned that choosing and gives back the old definitions of quasirandomness and regularity. In the previous section we saw an example when we needed different edge weights, but was identically one.

Let us consider a simple example in which has to take more than one value. Let be a star on vertices; that is, the vertex is adjacent to the vertices , and has degree 1 for . We let and for and choose . With these choices is easily seen to be a bipartite quasirandom; moreover, it is weighted regular.

A more sophisticated example relates weighted regularity to boundedness, which is the basic condition in the regularity lemma of Alon et al. [6]. Let us recall that is bounded with parameters , and is a function from to if for all , when , the inequality holds, where , and ; that is, is a “generalized edge density.” Then one can obtain an -regular partition if .

It is easy to check that the following graph is -quasi-random, in fact belongs to with appropriate weights, but is not bounded. Let , for . All edges between and are present; there is no edge between the sets and , while between and there is a random bipartite graph with edge probability . Of course, if is small enough compared to then cannot be bounded.

Similarly, one can show easily that whenever a graph is bounded, then with for all and appropriately defined edge weights is a dense subgraph of a graph which is -quasi-random. Hence, Theorem 6 can be applied for . We leave the details for the reader.

4.3. Natural Weighting of

Assume that . Let the vertex weight function be defined such that . We also assume that for every , as we did earlier in the paper. Then we define the natural weighting of the edges of ( can be replaced with other quasirandom graphs. Then the edge weights will be different. You can find more about this at the end of Section 4.5.) with respect to as follows: we let for all , . Observe that

We show that these weight functions determine a quasirandom weighting of . Let such that . Then

independent of the weights of and . Recalling the definition of quasirandomness it is easy to see that the natural weighting of is always quasirandom.

Note that natural weighting resembles Definition 1, where the lower bounds on and are dropped. It is closely related to the random model ; see, for example, [11]. Here is the expected degree sequence of with vertex set . The edges of are drawn independently, and the probability of including the edge is . Of course, the model is the special case of , and Lemma 7 holds without any conditions. The results in the remainder of the paper also hold in more general weightings; for simplicity, we work out the details for natural weighting.

Let be an arbitrary vertex and . Then the weighted degree of into in the graph is defined to be

where denotes the neighborhood of in the set . In particular the weighted degree of in is

We also have that

We define the weighted density of a weighted -regular pair to be

We have the following lemma.

Lemma 9. Let be a weighted -regular pair relative to the natural weighting of with weighted density . Let contain only such vertices that have weighted degree less than in the pair. Then .

Proof. Assume on the contrary that the set of “low-degree” vertices has a large weight. Observe that -regularity implies that

if . Using our assumption we get the following:

which is clearly a contradiction.

Let be disjoint subsets of , and assume that is a weighted -regular pair relative to a natural weighting of with weighted density at least for every . Set . Let be an integer constant, and assume that .

Lemma 10. Assume that with . Then there exist vertices such that

for every .

Proof. We find the vertices one by one. For we have that the weight of vertices of with weighted degree at most is at most using Lemma 9. Discard these low-degree vertices from ; then use the regularity condition again, this time for . We find that the weight of vertices having small degree into or is at most . Iterating this procedure we get that the weight of vertices that do not have large degree into at least one set is at most . Pick any of the large degree vertices from ; this is our choice for .

Next we repeat the process for finding , with the difference that we look for a vertex that has large degree into the sets for every . Since , the same procedure will work. Applying Lemma 9 we can find many vertices in such that the weighted degree of all of them into is at least for every . Pick any of these; this vertex is .

When it comes to finding we will work with the sets and for . Using induction it is easy to show that

for every . Since , we can iterate this procedure until we find all the vertices .

Assume now that there are clusters such that for all (here ) and all the pairs are weighted -regular relative to a natural weighting of with density at least . That is, we have a super-clique on clusters.

Next we construct the graph , a blown-up clique, as follows. First, we have disjoint -element set of vertices; this is the vertex set of . Then we connect any two vertices if they belong to different vertex sets. Before we state an embedding result, we need a simple lemma; the proof is left for the reader.

Lemma 11. Let be a weighted -regular pair with density , and for some let with and with . Then is a weighted -regular pair with and density .

We have the following embedding lemma.

Lemma 12. Let . If then .

Proof. First, apply Lemma 10 with and for . We find the vertices such that

Let ; then for every .

Next let and for . Using Lemma 11 we have that the new pairs are all weighted -regular with density at least . Hence, we can apply Lemma 10 again and find such that

for .

Continuing this process, in the th step we will work with the sets when applying Lemma 10. These sets are defined recursively as follows: and for every . Moreover, the pairs will be -regular with density at least for every .

In the last step, when , there are only two subclusters left, and . The pair will be weighted -regular with density at least . It is easy to find a (a complete bipartite graph) in this regular pair using Lemma 10. Clearly, we constructed the desired graph.

4.4. Illustration: A Weighted Version of the Erdős-Stone Theorem

Let be the number of edges in the Turán graph on vertices. That is, has the largest number of edges such that it does not contain a . It is well known that

The Erdős-Stone theorem states that if one has at least edges (where is a constant) in a graph on vertices then has a for any given natural number . In this section we show a weighted version. We take a natural weighting of and prove that if the total edge weight in is large then has a large blown-up clique. We remark that there are other results in the literature on the extremal theory of weighted graphs; see, for example, [13] by Bondy and Tuza and [14] by Füredi and Kündgen, although the setup of these papers is different from ours. Another version of the Erdős-Stone theorem for sparse graphs can be found in [15].

Theorem 13. For all integers and and every there exists an integer such that the following holds. Take the natural weighting of with vertex weight function and assume that . Let . If the total edge weight of is at least then contains as a subgraph.

Proof. We begin with applying the weighted regularity lemma with parameters and . We get an -regular partition with clusters . Let us construct the reduced graph as follows. The vertices of are identified by the nonexceptional clusters. We have an edge between two vertices of if the corresponding two clusters give an -regular pair with density at least . Hence, when we construct we lose edges of as follows: edges that are incident with some vertex of , edges that connect two vertices that belong to the same nonexceptional cluster, edges that are in some irregular pair, and edges that are in regular pairs with small density.

The outline of the proof is as follows. We will show that the loss in edge weight is small; hence, will have many edges. By Turán’s Theorem we will have a -clique in . Then we apply Lemma 12 and conclude the existence of a in .

(1)The total weight of edges that are incident with some vertex of can be estimated as follows:

(2)The nonexceptional clusters have weight . The total weight of edges inside nonexceptional clusters is at most

Since , we have that the total edge weight inside nonexceptional clusters is less than .

(3)Assume that is an irregular pair. Then

Since the number of irregular pairs is at most , we get that the total edge weight in irregular pairs is at most

(4)If the density of an -regular pair is small then we have the following inequality:

Since there can be at most pairs, the total edge weight in low density pairs is less than .

Putting together, we get that the total weight of edges that we lose when applying the weighted regularity lemma is at most . Hence, the total edge weight in the high-density regular pairs of is at least . The total weight of edges in a regular pair is . Assume that ; then the total edge weight would be at most . Since we have a larger edge weight in what is left after applying the regularity lemma, using Turán’s theorem we get that contains a . Every pair in this clique is a high-density -regular pair; hence, we can apply Lemma 12 and find the blown-up clique.

Remarks. One can arrive at the same conclusion perturbing the edge weights a little. Let be a fixed constant. Multiply the weight of the edge by any number . The resulting weighted graph will be quasirandom, and it is an easy exercise to show that one still has Theorem 13.

One can also show the weighted version of the Erdős-Stone-Simonovits theorem, a stability version of the above. Let be a family of forbidden subgraphs having chromatic number . Assume that the total edge weight in is close to , but does not contain some graph . Then the reduced graph cannot have a clique on vertices, but the number of edges in it will be close to . This implies that is close to a Turán graph and that in turn implies that the vertex set of can be partitioned into disjoint vertex classes in the following way: the vertex classes all have weight , the total weight of edges inside vertex classes is very small, and the weighted density of edges for every pair of disjoint classes is close to one.

4.5. Emphasized Sets

One cannot avoid having an exceptional cluster when applying the regularity lemma. That is, a linear number of vertices could be discarded in certain cases; a well-known example is the so-called half-graph. In general we do not have control on what is put into the exceptional cluster. However, using vertex weights one can at least partly control the set of discarded vertices. In what follows we show how to use the natural weighting of in order to have that the majority of some given emphasized set is put into nonexceptional clusters after applying the weighted regularity lemma, even if the set is of size . In fact we will do it for several emphasized sets at the same time. Notice that applying the usual regularity concept (even that of [6]) one may discard all vertices with small degrees.

Assume that is a fixed constant and is partitioned into the disjoint sets , and let . Further assume that as . Let for every . Define the following weighting of the vertices of : for we let

Observe that

thus, the total weight of the vertices is . Let and . The weight of the pair is

We showed above that equipped with such vertex and edge weights is a quasirandom graph. We call this particular weighting the natural weighting of with emphasized sets .

We can apply Theorem 6 for some relative to the natural weighting of . Choose so that . Since , we get that for all the majority of the vertices of are in nonexceptional clusters.

We remark that it is possible to define vertex weights not only for but also for much sparser quasirandom graphs when emphasizing subsets of . For example, assume that , and is partitioned into the disjoint sets . Then one will have the vertex weights of the above example, but the edge weights will be different:

whenever and . We leave the details for the reader.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank Péter Hajnal and Endre Szemerédi for the helpful discussions. This work was partially supported by the European Union and the European Social Fund through project FuturICT (Grant no.: TÁMOP-4.2.2.C-11/1/KONV-2012-0013). The first author was also supported by the ERC-AdG. 321104.

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