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E xtremal S ubgraphs of R andom G raphsPowerPoint Presentation

E xtremal S ubgraphs of R andom G raphs

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### Extremal Subgraphs ofRandom Graphs

Konstantinos Panagiotou

Institute ofTheoretical Computer Science

ETH Zürich

(joint work with Graham Brightwell and Angelika Steger)

Probabilistic Viewpoint

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 ?

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

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…

Could a similar picture be true for the random graph ?

- What we know…

Babai, Simonovits, Spencer (1990)

?

Question (Babai, Simonovits, Spencer):

Do we have

for all constant ? What if ?

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

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

“bipartite” with “ -partite”.

Our resultBrightwell, P., Steger

If then

regularity lemma

Only “few”

rededges

Properties of

Proof pipeline1

2

Even less

rededges

3

Properties of

maximum cuts

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)

regularity lemma

Only “few”

rededges

Properties of

Proof pipeline1

2

Even less

rededges

3

Properties of

maximum cuts

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“

2

Vertices which do

not fulfill the

“common neighb.”

property

“Too many” missing

edges:

• All „exceptional sets“ are small

• „Remainder“ is very sparse

• Bootstrapping

regularity lemma

Only “few”

rededges

Properties of

Proof pipeline1

2

Even less

rededges

3

Properties of

maximum cuts

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

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

- Let and be two bipartitions. Then
- Theorem. LetThe probability that has two optimal bipartitions and with is .

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

- 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

- 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)

- Increase of : number of edges added that go across
- Depends on the number of non-edges
across in

- Depends on the number of non-edges

- Hence

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

regularity lemma

Only “few”

rededges

Properties of

Proof pipeline1

2

Even less

rededges

3

Properties of

maximum cuts

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.

2

Properties of random graphs (for ):

- degree property
- degree property for subsets
- density property
(next three slides)

and all

except of vertices

Degree Property for Subsets- has the following property (aas):

we have

and

not fulfill the

“common neighb.”

property

Proof: Part - setup.2

common neighborhood (in ) too small

not fulfill the

“common neighb.”

property

Proof: Part - setup.2

“Too many” missing

edges:

Proof: Part - steps

2

- All „exceptional sets“ are small (next slide)
- „Remainder“ is very sparse (in two slides)
- Bootstrapping … (details omitted)

- General case:
- Inclusion/Exclusion
- We obtain a
- contradiction as
- soon as

- Missing edges:
- at least

Red graph is very sparse

- Suppose that we know more:
maxdeg

- Idea: remove all red edges, add all missing edges
- disjoint blue edges

in !

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 - continued

3

- Fix any optimal partition

- There are
disjoint triangles with and

...

- Problem: there might be exponentially many optimal bipartitions!

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.

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