The Effect of Induced Subgraphs

on

Quasi-randomness

Asaf Shapira & Raphael Yuster

Background

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

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

2. Proved that several “natural” properties guarantee that a

graph is p-quasi-random.

Abstract Question: What it means for a graph to be random?

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

to behave like a “truly” random one.

The CGW TheoremTheorem [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 of vertices U V spans ½p|U|2 edges

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

3. 1(G) pn and 2(G) = o(pn)

4. For any graph H on h vertices, G has nhpe(H) copies of H

5. G contains ½pn2 edges and p-4n4 copies of C4

6. All but o(n2) of the pairs u,v have p2n common neighbors

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

“” = (1+o(1))

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

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

[ADLRY ‘95], uses equivalence between regularity and

number of C4.

Relation to Extremal Combinatorics:

1. Central in the strong hypergraph generalizations of

Szemeredi’s regularity-lemma.

More on the CGW Theorem

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

Example: Having ½pn2 edges and p-3n3 copies of K3

is not a quasi-random property.

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

Recall that if we replace K3

with C4 we do get a quasi-random property.

No!

…but if we consider cuts of size (¼n,¾n) then the property is quasi-random.

Subgraph and Quasi-randomness

The effect of subgraphs on quasi-randomness:1. Having the correct number of edges + correct number of C4

is a quasi-random property. True also for any C2k.

2. Having the correct number of edges + correct number of K3

is not a quasi-random property. True for any non-bipartite H.

Question: Is it true that for any single H, the “distribution” of

copies of H affects the quasi-randomness of a graph?

[Simonovits and Sos ’97]: Yes. If all vertex sets U V

span |U|hpe(H) copies of H, then G is p-quasi-random.

Intutition: “Randomness is a hereditary property”.

Induced Subgraph and Quasi-Rand

Related Question: Can we expect to be able to deduce fromthe distribution of a single H that a graph is p-quasi-random.

[Simonovits and Sos ’97]: For any H, if all vertex sets U V

span |U|hpe(H) copies of H, then G is p-quasi-random.

Question: Is it true that for any single H, the “distribution” of

induced copies of H affects the quasi-randomness of a graph?

Answer: No. For any p and H, there is a q=q(p,H) for whichG(n,p) and G(n,q) behave identically w.r.t. induced copies of H.

Induced Subgraph and Quasi-Rand

Question: Is it true that for any single H, the “distribution” of induced copies of H affects the quasi-randomness of a graph?

[Simonovits and Sos ’03]: No. There are non quasi-random graphs, where all U V span |U|3p2(1-p) induced copies of P3 .

[Simonovits and Sos ’97]: For any H, if all vertex sets U V span |U|hpe(H) copies of H, then G is p-quasi-random.

Question: Is it true that for any H, if all vertex sets U V span |U|h H(p) induced copies of H, then G is quasi-random?

)1()(

2)(

)( ppHe

hHe

Hp

Definition:

A New Formulation

Lemma: Fix any H on h vertices. The following are equivalent:

1. G is such that all UV span |U|h H(p) induced copies of H.

2. G is such that all h-tuple U1,…,Uh of (arbitrary) size m span

h!mh H(p) induced copies of H with one vertex in each Ui . Observation: Consider any ordered h-tuple U1,…,Uh of

(arbitrary) size m in G(n,p). We actually expect U1,…,Uh to span mh H(p) induced copies of H with the vertex in Ui

playing the role of vertex vi of H.

Question: Is it true that for any H, if all vertex sets U V span |U|h H(p) induced copies of H, then G is quasi-random? NO

Definition: In that case, we say that U1,…,Uh have the correctnumber of induced embedded copies of H.

Main Result

Question: Is it true that for any single H, the “distribution” of induced

copies of H affects the quasi-randomness of a graph?

Theorem [S-Yuster ‘07]: Yes!

1. Assume all ordered h-tuples U1,…,Uh of (arbitrary) size m span the correct number of induced embedded copies of H. Then G is quasi-random.

2. In fact, G is either p-quasi-random or q-quasi-random.

Note: 1. We can’t expect to show that G is p-quasi-random. 2. We can’t consider only the number of induced copies of H in U1,…,Uh .

Proof Overview

Lemma [SS 91]: If G is composed of quasi-random graphswith the same density, then G itself is quasi-random. That is,

Suppose G is a k-partite graph on vertex sets V1,…,Vk, andmost of the bipartite graphs on (Vi,Vj) are p-quasi-random.Then G is also p-quasi-random.

Overall strategy: Show that G has a k-partition, where most of the quasi-random bipartite graphs have density p or most havedensity q.

Proof OverviewFact: If G is an h-partite graph on V1,…,Vh and all the bipartitegraphs (Vi,Vj) are quasi-random, then the number of inducedcopies of H is determined by the densities between (Vi,Vj).

denstiy = x1,3

density = x2,4

x1,2 · x1,3 · x1,4 · x2,3 · x2,4 · (1-x2,3)

V1

V3

V2

V4

The density of is

In fact even the number of induced embedded copies of H.

Proof Overview

We will show that in such a partition most densities are p or q.

[Szemeredi’s regularity lemma ‘79]: Any graph has a k-partition into V1,…,Vk s.t. most graphs on (Vi,Vj) are quasi-random. But not necessarily with the same density…

Assumption on G: We “know” the number of inducedembedded copies of H in each h-tuple of vertex sets V1…,Vh.

Fact: Given a partition where most bipartite graphs (Vi,Vj) arequasi-random, we “know” the number of induced copies of H.

We get (k)h polynomial equations relating the densities (Vi,Vj).

We will show that the only solution is p or q.

Proof Overview

Let W be a weighted complete graph on k vertices, with

weights 0 w(i,j) 1. For every mapping :[h][k] define

Key Lemma: Suppose that for all :[h][k] we have W()(p).

Then either most w(i,j) p or most w(i,j) q.

Notes: We cannot expect to show that most w(i,j) p.

Also, it is NOT true that either all w(i,j) p or all w(i,j) q.

1. w(i,j) stands for the density between Vi and Vj.2. W() (p) due to our assumption on G.

Proof Overview

Key Lemma: Suppose that for all :[h][k] we have

Proof idea: Introduce unknowns xij for each w(i,j).

Consider the system of polynomial equations:

Then either most w(i,j) p or most w(i,j) q.

First step: Show that the only solution is xij {p,q}.

Proof Overview

Proof idea: Suppose that for all :[h][k] we have

First step: Show that the only solution is xij {p,q}.

[Gottlieb ‘66]: If r k+h, then rank(A(r,h,k)) = .

Definition: For integers k h r, let A(r,h,k) be the inclusion

matrix of the k-element subsets of [r] and its h-element subsets.

k

r

h-element sets

k-element sets

AS,T = 1 iif ST

Proof Overview

3) Use Regularity-lemma + Packing results (Rodl or Wilson) to conclude that G is composed of quasi-random graphs with the same densities.

2) Some more (non-trivial) arguments needed to show that in fact most are p or most are q.

1) After (appropriate) manipulations this gives that the unique solution uses p and q.

Thank You