1 Introduction
Abstract

Quasi-random graphs can be informally described as graphs whose edge distribution closely resembles that of a truly random graph of the same edge density. Recently, Shapira and Yuster proved the following result on quasi-randomness of graphs. Let be a fixed integer, be positive reals satisfying and , and be a graph on vertices. If for every partition of the vertices of into sets of size , the number of complete graphs on vertices which have exactly one vertex in each of these sets is similar to what we would expect in a random graph, then the graph is quasi-random. However, the method of quasi-random hypergraphs they used did not provide enough information to resolve the case for graphs. In their work, Shapira and Yuster asked whether this case also forces the graph to be quasi-random. Janson also posed the same question in his study of quasi-randomness under the framework of graph limits. In this paper, we positively answer their question.

\title

Quasi-randomness of graph balanced cut properties \authorHao Huang \thanksDepartment of Mathematics, UCLA, Los Angeles, CA, 90095. Email: huanghao@math.ucla.edu. \andChoongbum Lee\thanksDepartment of Mathematics, UCLA, Los Angeles, CA 90095. Email: abdesire@math.ucla.edu. Research supported in part by Samsung Scholarship. \date

1 Introduction

The study of random structures has seen a tremendous success in modern combinatorics and theoretical computer science. One example is the Erdős-Rényi random graph proposed in the 1950’s and intensively studied thereafter. is the probability space of graphs over vertices where each pair of vertices forms an edge independently with probability . Random graphs are not only an interesting object of study on their own but also proved to be a powerful tool in solving numerous open problems. The success of random structures served as a natural motivation for the following question: How can one tell when a given structure behaves like a random one? Such structures are called quasi-random. In this paper we study quasi-random graphs, which, following Thomason [18, 19], can be informally defined as graphs whose edge distribution closely resembles that of a random graph (the formal definition will be given later). One fundamental result in the study of quasi-random graphs is the following theorem proved by Chung, Graham and Wilson [3] (here we only state part of their result).

Theorem 1.1

Fix a real . For an -vertex graph , define to be the number of edges in the induced subgraph spanned by vertex set , then the following properties are equivalent.

: For any subset of vertices , we have .

: For any subset of vertices of size , we have .

: and has cycles of length .

Throughout this paper, unless specified otherwise, when considering a subset of vertices such that for some , we tacitly assume that or . Since we mostly consider asymptotic values, this difference will not affect our calculation.

For a positive real , we say that a graph is -close to satisfying if for all , and similarly define it for other properties. The formal definition of equivalence of properties in Theorem 1.1 is as following: for every , there exists a such that if a graph is -close to satisfying one property, then it is -close to satisfying another.

We call a graph -quasi-random, or quasi-random if the density is clear from the context, if it satisfies , and consequently satisfies all of the equivalent properties of Theorem 1.1. We also say that a graph property is quasi-random if it is equivalent to . Note that the random graph with high probability is -quasi-random. However, it is not true that all the properties of random graphs are quasi-random. For example, it is easy to check that the property of having edges is not quasi-random (as an instance, there can be many isolated vertices). For more details on quasi-random graphs we refer the reader to the survey of Krivelevich and Sudakov [13]. Quasi-randomness was also studied in many other settings besides graphs, such as set systems [4], tournaments [5] and hypergraphs [6].

The main objective of our paper is to study the quasi-randomness of graph properties given by certain graph cuts. These kind of properties were first studied by Chung and Graham in [4, 7]. For a real , the cut property is the collection of graphs satisfying the following: for any of size , we have . As it turns out, for most values of , the cut property is quasi-random. In [4, 7], the authors proved the following beautiful theorem which characterizes the quasi-random cut properties.

Theorem 1.2

is quasi-random if and only if .

To see that is not quasi-random, Chung and Graham [4, 7] observed that the graph obtained by taking a random graph on vertices and an independent set on the remaining vertices, and then connecting these two graphs with a random bipartite graph with edge probability , satisfies but is not quasi-random.

A -cut is a partition of a vertex set into subsets , and if for a vector , the size of the sets satisfies for all , then we call this an -cut. An -cut is called balanced if for some , and is unbalanced otherwise. For a -uniform hypergraph and a cut of its vertex set, let be the number of hyperedges which have at most one vertex in each part for all .

A -uniform hypergraph is (weak) -quasi-random if for every subset of vertices , . Let be the following property: for every -cut , . Note that previously we mentioned the example which illustrate the non-quasi-randomness of . As noticed by Shapira and Yuster [15], a similar construction as above shows that is not quasi-random. In fact, they generalized Theorem 1.2 by proving the following theorem.

Theorem 1.3

Let be a positive integer. For -uniform hypergraphs, the cut property is quasi-random if and only if for some .

For a fixed graph , let be the following property : for every subset , the number of copies of in is . In [16], Simonovits and Sós proved that is equivalent to and hence is quasi-random. For a fixed graph , as a common generalization of Chung and Graham’s and Simonovits and Sós’ theorems, we can consider the number of copies having one vertex in each part of a cut. Let us consider the cases when is a clique of size .

Definition 1.4

Let be positive integers such that , and let be a vector of positive real numbers satisfying . We say that a graph satisfies the cut property if for every -cut , the number of copies of which have at most one vertex in each of the sets is .

Shapira and Yuster [15] proved that for , is quasi-random if is unbalanced (note that is quasi-random if and only if is unbalanced). This result is a corollary of Theorem 1.3 by the following argument. For a graph satisfying , consider the -uniform hypergraph on the same vertex set where a -tuple of vertices forms an hyperedge if and only if they form a clique in . Then satisfies and thus is quasi-random. By the definition of the quasi-randomness of hypergraphs, this in turn implies that the number of cliques of size inside every subset of is “correct”, and thus by Simonovits and Sós’ result, is quasi-random.

Note that for balanced this approach does not give enough information, since it is not clear if there exists a graph whose hypergraph constructed by the above mentioned process is not quasi-random but satisfies (nonetheless as the reader might suspect, the properties and are closely related even for balanced ). Shapira and Yuster made this observation and left the balanced case as an open question asking whether it is quasi-random or not (in fact, they asked the question for , but here we consider the slightly more general question for all balanced as mentioned above). Janson [11] independently posed the same question in his paper that studied quasi-randomness under the framework of graph limits. In this paper, we settle this question by proving the following theorem :

Theorem 1.5

Fix a real and positive integers such that . For every positive , there exists a positive such that the following is true. If is a graph which has density and is -close to satisfying the balanced cut property , then is -close to being -quasi-random.

The rest of the paper is organized as follows. In Section 2 we introduce the notations we are going to use throughout the paper and state previously known results that we need later. In Section 3 we give a detailed proof of the most important base case of Theorem 1.5, triangle balanced cut property, i.e. . In Section 4, we prove the general case as a consequence of the base case. The last section contains some concluding remarks and open problems for further study.

2 Preliminaries

Given a graph and two vertex sets , we denote by the set of edges which have one end point in and the other in . Also we write to indicate the number of edges and for the density. For a cut of the vertex set, we say that a triangle with vertices crosses the cut if it contains at most one vertex from each set, and denote it by . We use for the number of triangles with vertices . For a -uniform hypergraph and a partition of its vertex set into parts, we define its density vector as the vector in indexed by the -subsets of whose -entry is the density of hyperedges which have exactly one vertex in each of the sets . Throughout the paper, we always use subscripts such as to indicate that the parameter comes from Theorem 2.6.

To state asymptotic results, we utilize the following standard notations. For two positive-valued functions and , write if there exists a positive constant such that , if . Also, if there exists a positive constant such that .

To isolate the unnecessary complication arising from the error terms, we will use the notation if and say that are -equal. For two vectors, we define if . We omit the proof of the following simple properties (we implicitly assume that the following operations are performed a constant number of times in total). Let and be positive constants.

(1a) (Finite transitivity) If and , then .
(1b) (Complete transitivity) For a finite set of numbers . If for every , then there exists such that for all .
(2) (Additivity) If and , then .
(3) (Scalar product) If and , then and .
(4) (Product) If are bounded above by , then and implies that .
(5) (Square root) If both and are greater than , then implies that .
(6) For the linear equation , if all the entries of an invertible matrix are bounded by , and the determinant of is bounded from below by , then .
(7) If , then either or .

2.1 Extremal Graph Theory

To prove the main theorem, we use the regularity lemma developed by Szemerédi [17]. Let be a graph and be fixed. A disjoint pair of sets is called an -regular pair if such that satisfies . A vertex partition is called an -regular partition if (i) the sizes of differ by at most 1, and (ii) is -regular for all but at most pairs . The regularity lemma states that every large enough graph admits a regular partition. In our proof, we use a slightly different form which can be found in [12]:

Theorem 2.1 (Regularity Lemma)

For every real and positive integers there exists constants and such that given any , the vertex set of any -vertex graph can be partitioned into sets for some divisible by and satisfying , so that

  • for every .

  • for all .

  • Construct a reduced graph on vertices such that in if and only if is -regular in . Then the reduced graph has minimum degree at least .

As one can see in the following lemma, regular pairs are useful in counting small subgraphs of a graph (this lemma can easily be generalized to other subgraphs).

Lemma 2.2

Let be subsets of vertices. If the pair is -regular with density for every distinct , then the number of triangles is

Proof.  If a vertex has degree in and in , then by the regularity of the pair , there will be triangles which contain the vertex . By the regularity of the pair , there are at least vertices in which have at least neighbors in , and similar holds for the pair . Hence there are at least such vertices satisfying both conditions. Moreover, since each vertex in is contained in at most triangles, there are at most triangles which do not contain such vertex from . Therefore we have,

For a fixed graph , a perfect -factor of a large graph is a collection of vertex disjoint copies of that cover all the vertices of . The next theorem is a classical theorem proved by Hajnal and Szemerédi [9] which establishes a sufficient minimum degree condition for the existence of a perfect clique factor.

Theorem 2.3 ([9])

Let be a fixed positive integer and be divisible by . If is a graph on vertices with minimum degree at least , then contains a perfect -factor.

2.2 Concentration

The following concentration result of Hoeffding [10] and Azuma [2] will be used several times during the proof (see also [14, Theorem 3.10]).

Theorem 2.4 (Hoeffding-Azuma Inequality)

Let be constants, and let be a martingale difference sequence with for each . Then for any ,

The next lemma is a corollary of Hoeffding-Azuma’s inequality.

Lemma 2.5

Let be a graph with and for some fixed real . Let be a random subset of constructed by selecting every vertex independently with probability . Then with probability at least .

Proof.  Arbitrarily label the vertices by and consider the vertex exposure martingale. More precisely, let be the number of edges within incident to among the vertices ( if ), and note that . Also note that forms a martingale such that for all . Thus by Hoeffding-Azuma’s inequality (Theorem 2.4),

Since , by selecting , we obtain with probability at least (see, e.g., [1, Theorem 7.2.3] for more on vertex exposure martingales).

Note that the probability of success in this lemma can be improved by carefully choosing our parameters. However, Lemma 2.5 as stated is already strong enough for our later applications.

2.3 Quasi-randomness of hypergraph cut properties

Recall the cut property defined in the introduction, and the fact that it is closely related to the clique cut property . While proving Theorem 1.3, Shapira and Yuster also characterized the structure of hypergraphs which satisfy the balanced cut property . Let be fixed and be an integer. In order to classify the -uniform hypergraphs satisfying the balanced cut property, we first look at certain edge-weighted hypergraphs. Fix a set of size , and consider the weighted hypergraph on the vertex set such that the hyperedge has density for all . Let be the vector in representing this weighted hypergraph (each coordinate corresponds to a -subset of , and the value of the vector at the coordinate is the edge-weight of that hyperedge), and let be the affine subspace of spanned by the vectors for all possible sets of size . In [15], the authors proved that the structure of a (non-weighted) hypergraph which is -close to satisfying the balanced cut property can be described by the vector space (note that the vector which has constant weight lies in this space).

Theorem 2.6 ([15])

Let be fixed. There exists a real such that for every , and for every divisible by 111The authors omitted the divisibility condition in their paper [15]., there exists so that the following holds. If is a -uniform hypergraph with density which is -close to satisfying the balanced cut property , then for any partition of into equal parts, the density vector of this partition satisfies for some vector .

A part of the proof of Shapira and Yuster’s theorem relies on showing that certain matrices have full rank, and they establish this result by using the following famous result from algebraic combinatorics proved by Gottlieb [8]. For a finite set and integers and satisfying , denote by the versus inclusion matrix of which is the - matrix whose rows are indexed by the -element subsets of , columns are indexed by the -elements subsets of , and entry is 1 if and only if .

Theorem 2.7

for all .

3 Base case - Triangle Balanced Cut

In this section we prove a special case, triangle balanced cut property, of the main theorem. Our proof consists of several steps. Let be a graph which satisfies the triangle balanced cut property. First we apply the regularity lemma to describe the structure of by an -regular partition . This step allows us to count the edges or triangles effectively using regularity of the pairs. From this point on, we focus only on the cuts whose parts consist of a union of the sets . In the next step, we swap some vertices of and . By the triangle cut property, we can obtain an algebraic relation of the densities inside and between and . After doing this, the problem is transformed into solving a system of nonlinear equations, which basically implies that inside any clique of the reduced graph, most of the densities are very close to each other. Finally resorting to results from extremal graph theory, we can conclude that almost all the densities are equal and thus prove the quasi-randomness of triangle balanced cut property.

To show that our given graph is quasi-randomn, ideally, we would like to show that the densities of edges between pair of parts in the regular partition is (almost) equal to each other. However, instead of establishing quasi-randomness through verifying this strong condition, we will derive it from a slightly weaker condition. More specifically, we will use the fact that if in an -regular partition of the graph, the density of edges in most of the pairs of parts are equal to each other, then the graph is quasi-random (there are some dependencies in parameters). Following is the main theorem of this section.

Theorem 3.1

Fix a real and an integer . For every positive , there exists a positive real such that the following is true. If is a graph which has density and is -close to satisfying the triangle balanced cut property , then is -close to being -quasi-random.

Let be a graph -close to satisfying . By applying the regularity lemma, Theorem 2.1, to , we get an -regular equipartition . We can assume that by deleting at most vertices. The reason this can be done is that later when we use the triangle cut property to count the number of triangles, the error term that this deletion creates is at most which is negligible comparing to when is sufficiently large. Also in the definition of quasi-randomness, the error term from counting edges is at most , which is also .

Now denote the edge density within by , the edge density between and by , and the density of triangles in the tripartite graph formed by by . Call a triple regular if each of the three pairs is regular.

Consider a family of partitions of given as follows:

In other words, we pick and both containing -proportion of vertices in and uniformly at random and exchange them to form a new equipartition . To be precise, for fixed , the notation represents a family of random partitions and not necessarily an individual partition. For convenience we assume that is a partition constructed as above which satisfies some explicit properties that we soon mention which a.a.s. hold for random partitions. Denote the new triangle density vector of by .

Note that every -cut of the index set also gives a -cut of . With a slight abuse of notation, we use to indicate that , and completely belongs to different parts of the cut induced by .

By the triangle balanced cut property, for every ,

So . Let be the matrix whose rows are indexed by the -cuts of the vertex set and columns are indexed by the triples , where the -entry of is if and only if . The observation above implies where is the all-one vector. Thus if we let , then . From this equation we hope to get useful information about the densities and . With the help of the following lemma, we can compute the new densities , and thus the modified density vector , in terms of the densities and .

Lemma 3.2

Let satisfy for every and assume that the graph is large enough. Then for all , there exists a choice of sets such that the following holds.

If is a regular triple, then

Let . Then

Moreover, for the case and , if is a regular triple, then

Proof.  Throughout the proof, we rely on the fact that some events hold with probability . Since there are fixed number of events involved, without further mentioning, we will assume that all the involved events happen together at the same time.

The claim clearly holds for the cases and .

For , if , then the density is not affected by the swap of vertices in and so it remains the same with . In the case that , without loss of generality we assume and . We also assume that there are triangles with a fixed vertex and two other vertices belonging to and respectively (note that ). After swapping subset with such that , we know that the number of triangles in triple changes by .

Assume , instead of taking vertices uniformly at random, take every vertex in (or ) independently with probability . This gives random variables for having Bernoulli distribution with parameter . Let and . These random variables represent the number of vertices chosen for , and the number of triangles in the triple that contain these chosen vertices, respectively. It is easy to see

and by Hoeffding-Azuma’s inequality (Theorem 2.4)

Let , and the second probability decreases much faster than the first probability, thus we know that conditioned on the event , is also concentrated at its expectation . From here we know the number of triangles changes by

Therefore the new density is

We can use a similar method to compute when .

Let , be as in (1), and let , . Then we have the identity

Since , the triples and are regular. Thus by Lemma 2.2,

To compute , let be the collection of edges such that their end points have common neighbors in . By the regularity of the pair , there are at most vertices in which do not have neighbors in , otherwise taking this set of vertices and will contradict the regularity. If is not such a vertex, then since by the hypothesis , by using the regularity, we see that there are at most other vertices in which do not have common neighbors with . Consequently there are at most edges inside which do not have common neighbors inside . We call these edges “exceptional”. Recall that denotes the density of edges in , thus . By Lemma 2.5 and the calculation from part (1) there exists a choice of of size such that,

for all . Note that the number of triangles can be computed by adding the number of triangles containing the edges in and then the number of triangles containing the “exceptional” edges (recall that there are at most of the such edges). The latter can be crudely bounded by . Since each edge in is contained in triangles (within the triple ),

Similarly we can show that there exists a choice of of size that gives

for all . Combining all the results together, we can conclude the existence of sets , such that

Part is just a straightforward computation from the definition of and .

Lemma 3.3

Let satisfy for every and assume that the graph is large enough. If is a regular triple, then