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E xtremal S ubgraphs of R andom G raphs. Konstantinos Panagiotou Institute of Theoretical Computer Science ETH Zürich. (joint work with Graham Brightwell and Angelika Steger ). Maximum bipartite subgraph. Motivation. Maximum triangle-free subgraph. Maximum bipartite subgraph.

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E xtremal s ubgraphs of r andom g raphs

Extremal Subgraphs ofRandom Graphs

Konstantinos Panagiotou

Institute ofTheoretical Computer Science

ETH Zürich

(joint work with Graham Brightwell and Angelika Steger)


Motivation

Maximum

bipartite

subgraph

Motivation

Maximum

triangle-free

subgraph


Maximum

bipartite

subgraph

But…

Maximum

triangle-free

subgraph


This Talk

Probabilistic Viewpoint


Random graph
Random Graph .

set of maximum triangle-free subgraphs

set of maximum bipartite subgraphs

size of a maximum triangle-free subgraph

size of a maximum bipartite subgraph

Questions:

  • ?

  • If so, do we also have ?


Random graph1
Random Graph .

here: [Mantel 1907]

we know from random graph theory:

is aas. a forest

hence: aas.

?


Easy to see:

Necessary Condition:

no edge within the larger class can be added (without creating a triangle)

Pr[edge can be added]

Pr[no edge can be added]

If then



A related problem
A Related Problem

Erdös, Kleitman, Rothschild (1976)

Almost all large triangle-freegraphs are bipartite.

randomtriangle-freegraph

on verticeswithedges

Prömel, Steger (1996), Steger (2005),

Osthus, Prömel, Taraz (2003):


Pictures say more
Pictures say more…

Could a similar picture be true for the random graph ?


What we know
- What we know…

Babai, Simonovits, Spencer (1990)

?

Question (Babai, Simonovits, Spencer):

Do we have

for all constant ? What if ?


Our result

In fact, the theorem is true for all ³2 if we

replace “triangle-free” with “ -free” and

“bipartite” with “ -partite”.

Our result

Brightwell, P., Steger

If then


Proof pipeline

Szemerédi’s

regularity lemma

Only “few”

rededges

Properties of

Proof pipeline

1

2

Even less

rededges

3

Properties of

maximum cuts


Proof part
Proof: Part .

1

Szemerédi’s Regularity

Lemma (sparse case)

Kohayakawa (1997)

Rödl (2003)

Pair is regular

anddense

(Probabilistic)

Embedding Lemma

Gerke, Marciniszyn,

Steger (2005)

Contradiction!

(Embedding Lemma)


Proof pipeline1

Szemeredi’s

regularity lemma

Only “few”

rededges

Properties of

Proof pipeline

1

2

Even less

rededges

3

Properties of

maximum cuts


Proof part1
Proof: Part .

2

  • Start with a „best“ bipartition

    • „best“: the number of red edges is minimized

  • Main idea: if we can exhibit more „missing“ edges than red edges, then we achieve a contradiction

„missing“


Proof: Part - setup.

2

Vertices which do

not fulfill the

“common neighb.”

property

“Too many” missing

edges:

• All „exceptional sets“ are small

• „Remainder“ is very sparse

• Bootstrapping


Proof pipeline2

Szemeredi’s

regularity lemma

Only “few”

rededges

Properties of

Proof pipeline

1

2

Even less

rededges

3

Properties of

maximum cuts


Proof part2
Proof: Part .

3

How does a maximum triangle-free subgraph - of look like?

is a bipartition

red edges

Observe: has to contain all edges of

between except possibly some edges

has to be a partition that is within to

an optimum partition


Proof part continued
Proof: Part - continued

3

Howlikelyisitthatonecanadd a singlerededge after removing

atmostedgesfrom (withoutcreating a triangle)?

  • There are

    disjoint triangles with and

...

  • Hence

Need: number of partitions to be considered


On the number of optimal bipartitions
On the number of optimal bipartitions

  • Let and be two bipartitions. Then

  • Theorem. LetThe probability that has two optimal bipartitions and with is .


Sketch of proof
Sketch of proof

  • Let denote the size of max-cut

  • Consider the expected change

    Two points of view

    • Delete edge from

    • Add edge to bring desired probability into play


Deleting edges
Deleting edges

  • If we delete edges, the max-cut decreases by at most the number of edges which are removed from all max-cuts

  • Upper bound: consider a fixed max-cut

  • Hence


Adding edges
Adding edges

  • Fix a max-cut in

  • When adding edges, one of two cases occurs:

    • remains (one of) the max-cuts

    • A different max-cut “overtakes”

  • Hence

    and


Adding edges ii
Adding edges (II)

  • Increase of : number of edges added that go across

    • Depends on the number of non-edges

      across in

  • Hence


Adding edges iii
Adding edges (III)

  • Suppose “overtakes”

  • Also:

  • If the event occurs, then there were added more edges to than to

  • It turns out

  • Putting the upper & lower bounds together yields the theorem


Proof pipeline3

Szemeredi’s

regularity lemma

Only “few”

rededges

Properties of

Proof pipeline

1

2

Even less

rededges

3

Properties of

maximum cuts



Summary
Summary

  • We showed that for sufficiently dense random graphs the maximum triangle-free and maximum bipartite subgraphs are identical.

  • On the way we obtained information about the structure of the max-cut in random graphs.

  • Our results generalize to arbitrary (constant-size) cliques.



Warm-up for Part .

2

Properties of random graphs (for ):

  • degree property

  • degree property for subsets

  • density property

    (next three slides)


Degree property

For all

Degree Property

  • has the following property (aas):

we have

and


Degree property for subsets

For all

and all

except of vertices

Degree Property for Subsets

  • has the following property (aas):

we have

and


Density property

with

we have

“Density” property

  • has the following property (aas):

For all ,


Proof part setup

Vertices which do

not fulfill the

“common neighb.”

property

Proof: Part - setup.

2

common neighborhood (in ) too small


Proof part setup1

Vertices which do

not fulfill the

“common neighb.”

property

Proof: Part - setup.

2

“Too many” missing

edges:


Proof part steps
Proof: Part - steps

2

  • All „exceptional sets“ are small (next slide)

  • „Remainder“ is very sparse (in two slides)

  • Bootstrapping … (details omitted)



Example is small1

  • General case:

  • Inclusion/Exclusion

  • We obtain a

  • contradiction as

  • soon as

Example: is small

  • Missing edges:

  • at least


Red graph is very sparse
Red graph is very sparse

  • Suppose that we know more:

    maxdeg

  • Idea: remove all red edges, add all missing edges

  • disjoint blue edges

in !


Proof part3

…and is an optimal

bipartition!

We say that has gap 1.

In general: red edges: has gap

Remark: !

Proof: Part .

3

  • Suppose

  • How does a maximum triangle-free subgraph look like?

up to one edge!


Proof part continued1

Proof: Part - continued

3

  • Fix any optimal partition

  • There are

    disjoint triangles with and

...

  • Problem: there might be exponentially many optimal bipartitions!


Problem erd s 1976
Problem - Erdös (1976)

Conjecture (Erdös 1976):

Everytriangle-free graph on n vertices can be made bipartite by deleting at most n2/25 edges.

Note:

If true, then best possible.