The Quasi-Randomness

of

Hypergraph Cut Properties

Asaf Shapira & Raphael Yuster

Background

Chung, Graham, and Wilson ’89, Thomason ‘87: 1. Defined the notion of a p-quasi-random graph = A graph

that “behaves” like a typical graph generated by G(n,p).

2. Proved that several “natural” properties that hold in G(n,p)

whp “force” a graph to be p-quasi-random.

Abstract Question: When can we say that a single graph

behaves like a random graph?

“Concrete” problem: Which graph properties “force” a graph

to behave like a “truly” random one.

The CGW Theorem

Theorem [CGW ‘89]: Fix any 0<p<1, and let G=(V,E) be a

graph on n vertices. The following are equivalent:

1. Any set U V spans ½p|U|2 edges

2. Any set U V of size ½n spans ½p|U|2 edges

3. G contains ½pn2 edges and p4n4 copies of C4

4. Most pairs u,v have co-degree p2n

Definition: A graph that satisfies any (and therefore all) theabove properties is p-quasi-random, or just quasi-random.

Note: All the above hold whp in G(n,p).

Quasi-Random Properties

Definition: Say that a graph property is quasi-random if it isequivalent to the properties in the CGW theorem.

“The” Question: Which graph properties are quasi-random?

Any (reasonable) property that holds in G(n,p) whp?

Example 1: Having ½pn2 edges and p3n3 copies of K3

is not a quasi-random property.

Recall that if we replace K3

with C4 we do get a quasi-random property.

No!

Example 2: Having degrees pn is not a quasi-random prop.

…but having co-degrees p2n is a quasi-random property.

The Chung-Graham Theorem

Theorem [Chung-Graham ’89]:

1. Having ¼ pn2 edges crossing all cuts of size (½n,½n) is not a quasi-random property.

2. For any 0<<½ , having (1-)pn2 edges crossing all cuts of size (n,(1-)n) is a quasi-random property.

[CG ‘89] Gave two proofs of (2). One using a counting argument, and another algebraic proof based on the rank of certain intersection matrices.

To get (1), take an Independent set on n/2 vertices, a cliqueon the rest, and connect them with a random graph.

[Janson ‘09] Gave another proofs of (2), using graph limits.

Quasi-Random Hypergraphs

What is a quasi-random hypergraph?

Answer 1: The “obvious” generalization of quasi-random

graphs. Every set of vertices has the “correct” edge density.

Definition: This is called “weak” quasi-randomness.

Why? Because it does not imply certain things that are

implied by quasi-randomness in graphs.

Answer 2: “Strong” quasi-randomness.

Fact: Strong Quasi-Randomness Weak Quasi-Randomness

Notation: “P is Quasi-Random” means P Weak Quasi-Rand

Our Main Results

Theorem 1 [S-Yuster ’09]:

1. If = (1/k,…,1/k) then P is not quasi-random.

Definition: Let =(1,…,k) satisfy 0<i<1 and I =1.

Let P be the following property of k-uniform hypergraphs:

Any (1n,…,kn)-cut has the correct number of edges

crossing it.

2. If (1/k,…,1/k) then P is quasi-random.

Theorem 2 [S-Yuster ’09]:1. When = (½,½) the only way a non-quasi random graph can satisfy P is the “trivial” one.2. Same result conditionally holds in hypergraphs.

Proof Overview

Theorem: If = (1/k,…,1/k) then P is not quasi-random.

Definition: Let =(1,…,k) satisfy 0<i<1 and I =1. We let P be the

following property of k-uniform hypergraphs: Any (1n,…,kn)-cut has the

correct number of edges crossing it.

0 2pp

[CG‘89] For k=2:

For arbitrary k2:

i verticesk-i vertices2ip/k

Proof OverviewTheorem: If (1/k,…,1/k) then P is not quasi-random.

Proof (of k=2): It is enough to show that every set of verticesof size n has the correct edge density. Let A be such a set.

p2p1 p3

Let 0c, and “re-shuffle” the partition by randomly picking cn vertices from A and (-c)n vertices from V-A.

1. We know the expected number of edges in the new cut.

2. This expectation is a linear function in p1 , p2 , p3.

3. Using c{0,,/2} we get 3 linear equations, which have a unique solution p1=p2=p3=p when 1/2.

A |A|=n

V-A |V-A|=(1-)n

Proof OverviewTheorem 2: When = (½,½) the only way a non-quasi random graph can satisfy P is the “trivial” one.

Instead of thinking about graphs, let’s consider the problemof assigning weights to the edges of the complete graph,s.t. for any (n/2,n/2)-cut, the total weight crossing it is p.

p

The random graph

0 2pp

The example showing that P is not quasi random

What is “trivial”? Two ways a graph can satisfy P

There are many such solutions

Proof Overview

What is “trivial”? Two ways a graph can satisfy P

p

The random graph

(1)

0 2pp

The example showing that P is not quasi random

(2)Satisfying P is equivalent to satisfying a set of linear equations:

1. Unknowns are the weights of the edges.

2. We have one linear equation for any (n/2,n/2)-cut

Definition: A trivial solution is any affine combination of solution (1) and (a collection of) solution (2).

Proof Overview

p

The random graph

(1)

0 2pp

The example showing that P is not quasi random

(2)

Definition: A trivial solution is any affine combination of solution (1) and (a collection of) solution (2).

Theorem 2: When = (½,½) the only solutions satisfying P are the trivial ones.

Proof Overview

Definition: A trivial solution is any affine combination of solution (1) and (2).

Theorem 2: When = (½,½) the only solutions satisfying P are the trivial ones.

Proof: Any solution is a solution of the linear system Ax=p.

Recall Satisfying P is equivalent to satisfying a set of linear equations: 1. Unknowns are the weights of the edges. 2. We have one equation for any (n/2,n/2)-cut

p The random graph (1)

0 2pp The example showing that

P is not quasi random

(2)

Step 2: span[solutions (2) – (1)] has dimension n-1.

Step 1: rank(A)

Definition: Let us write this as Ax=p.

Note: A is an matrix.

Proof Overview

Definition: A trivial solution is any affine combination of solution (1) and (2).

p The random graph (1)

0 2pp The example showing that

P is not quasi random

(2)

Step 2: trivial solutions have dimension n-1.

Proof: For every solution (2) consider the vector of pairs (v1,vi). 2p0

p

n/2-1 of the entries are 2p, the other are p.

After subtracting (1) from these vectors, we get, for everysubset S [n-1] of size n/2, a vector vS, satisfying:

1. vS(i) = 0 if i S.2. vS(i) = p if i S .

This collection spans Rn-1

Proof Overview

Note: A is an matrix

n/2 n/2n/2 n/2-1

Proof: Take the vector vS corresponding to some cut.

vS,t = vector of cut obtained by moving t from S to V-S.

We first prove that matrix of (n/2,n/2-1)-cuts has full rank.

(vS,t – vS) 1-1 t

(vS,t – vS) tvs-

S V-S

C-c

Step 1: rank(A)

Proof Overview

Conclusion: A spans the rows of the matrix I(2,n/2,n-1)

n/2-element subsets of [n-1]

2-element subsets of [n-1]

IS,T = 1 iif ST

[Gottlieb ‘66]: rank(I(2,h,k)) = .

Step 1: rank(A)

Concluding Remarks

Coro: If (1/3,1/3,1/3) then P is quasi-random

Definition: Let =(1,2,3) satisfy 0<i<1 and i =1.

Let P be the following graph property:

Any (1n,2n,3n)-cut is crossed by the “correct” number of K3.

Open Problem: What happens when = (1/3,1/3,1/3)?

Proof: Replace every K3 with a 3-hyper edge. We geta hypergraph satisfying P, which must be quasi-random byTheorem 1. This means that in the graph, any set of verticeshas the “correct” number of K3. A theorem of Simonovits-Sosimplies that the graph must be quasi-random.

Thank You

Background

Relation to (theoretical) computer science:

1. Conditions of randomness that are verifiable in polynomial

time. For example, using number of C4, or using 2(G).

2. Algorithmic version of Szemeredi’s regularity-lemma:

[Alon et al. ’95] Uses equivalence between quasi-randomenss

and co-degrees.

Background

Relation to Extremal Combinatorics:

1. Central in the strong hypergraph generalizations of

Szemeredi’s regularity-lemma [RSSN’04, Gowers’06, Tao’06].

Quasi-Random Groups [Gowers ‘07]

Generalized Quasi-Random Graphs [Lovasz-Sos ‘06]

Quasi-Random Set Systems [Chung-Graham ‘91]