Small subgraphs in the Achlioptas process

Reto Spöhel, ETH ZürichJoint work with Torsten Mütze and Henning Thomas

Introduction

• Goal: Avoid creating a copy of some fixed graph F

• Achlioptas process:

• start with the empty graph on n vertices

• in each step r edges are chosen uniformly at random (among all edges never seen before)

• select one of the r edges that is inserted into the graph, the remaining r – 1 edges are discarded

How long can we avoid F by this freedom of choice?

F = , r = 2

Introduction• N0=N0(F, r, n) is a threshold:

N0=N0(F, r, n) N = N(n) = # steps

There is a strategy that avoids

creating a copy of F withprobability 1-o(1)

N /N0

If F is a cycle, a clique or a complete bipartite graph with parts of equal

size, an explicit threshold function is known. (Krivelevich, Loh, Sudakov, 2007+)

Every strategy will be forced

to create a copy of F withprobability 1-o(1)

N [N0

n1.2

F = , r=2

n1.286…

r=3n1.333…

r=4

n1

r=1

Our Result

• Theorem:Let F be any fixed graph and let r R 2 be a fixed integer. Then the threshold for avoiding F in the Achlioptas process with parameter r is

where

• We solve the problem in full generality.

In the following we will try to convey an

intuition for the threshold formula.

r-matched graph r=2

Graph

r-matched Graphs

Random graph Random r-matched graph- generate Gn,m

- randomly partition the m edges into sets of size r

a family of disjoint r-sets of edges.

- have well-defined notion of subgraph inclusion

- Achlioptas process after N steps is distributed as

The Gluing Intuition

‚ gluingtogether’

E[ # copies of in ]

F

F 0 ³ E[ # copies of F in Gn,m ]

Threshold for „ contains “

= Threshold for „ Gn,m contains F “

F 0

³ nv(F) (m/n2)e(F)

Bollobás (1981): Threshold for the appearance of F in Gn,m is given by

m=m(n)

The Gluing Intuition

Example:

F = , r=2

Greedy strategy:

e/v = 5/4

As long asa.a.s. this subgraph does not appear, and hence we do not lose.

Our approach: view Achlioptas process as ‘static’ object and use (the r-matched version of) the above theorem.

1st Observation: Subgraph Sequences

F = , r=2

e/v = 11/8 = 1.375

Greedy strategy:

0 Maximization over a sequence of subgraphs of F

e/v = 15/10 = 1.5

Optimal strategy:

As long as a.a.s. this subgraph does not appear, and hence we do not lose.

2nd Observation: Edge OrderingsOrdered graph: pair

oldest edge

youngest edge2

3

7

4

6

1

5

0 Minimization over all possible edge orderings of F

1

2

2

e/v = 19/14 = 1.357...

2

F = , r=2

Edge ordering ¼1:

1

2

Optimal Strategy for ¼1:

12

2

12

e/v = 17/12 = 1.417...

Edge ordering ¼2:

12

12

2Optimal Strategy for ¼2:

F3-

F3-

F3-

F3-

F3-

F3-

F3-

F3-

4

4

4

4

F2-

F2-

F2-

F2-

3

4F1-3 F1-

r-1

4

Calculating the ThresholdMinimize over all possible edge orderings ¼ of FMaximize e(J)/v(J) over all

subgraphs J4F ¼

r2

34

F2

34

F 1

F ¼

Maximization over a sequence of subgraphs of F

5

J

H1

H2

H3

H1

H2H3

H3

H3

Calculating the Threshold (Example)

7

F6-…

F ¼

1

3 4

6

2

5

34

65

2

77

F = , r=2

Minimize over all possible edge orderings ¼ of FMaximize e(J)/v(J) over all

subgraphs J4F ¼5

3

7

4

6 5

3

7

4

65

7

7 7

7J

e(J)/v(J) = 19/14

Maximization over a sequence of subgraphs of F

1

3

7

4

6

2

5

F

(F, ¼) 23

7

4

6 5

F1-

3

7

4

6 5

F2-

Maximization over a sequence of subgraphs of F

Our Result ( explained)

• Theorem:Let F be any fixed graph and let r R 2 be a fixed integer. Then the threshold for avoiding F in the Achlioptas process with parameter r is

where

Minimization over all possible edge orderings of FMaximize e(J)/v(J) over all

subgraphs J4F ¼

A Canonical Strategy

• In order to avoid copies of F, we want to keep the number of copies of all (ordered!) subgraphs (H, ¼) of F as low as possible.

• For N = n2 – µ fixed:

• The problem has a recursive structure.

• We can compute a canonical strategy and values ¸µ(H, ¼) such that in N = n2 – µ many steps, at most copies of (H, ¼) are created (for all subgraphs (H, ¼) simultaneously!)

• In order to avoid copies of F we use this canonical strategy with

• Alternatively, the threshold can be defined directly via the recursion.

About the ¸-values

21

harm

les

sdangero

us

F = , r=2

¸µ(H, ¼) is the exponent of the expected number of copies of a ‚typical‘ history graph J of (H, ¼),

A Canonical Strategy

• For each edge calculate the level of danger it entails as the most dangerous (ordered) subgraph this edge would close

• Among all edges, pick the least dangerous one

f1 f2 fr…

is more dangerous than

5

Lower Bound ProofDeterministic Lemma: By our strategy, each black copy of some ordered graph is contained in a copy of a grey-black r-matched graph H’ with expectation at most . . .

1

2

H’“History graph”

• If , each possible history does a.a.s. not appear in .

• Constantly many histories H’of ending up with a copy of F

a.a.s. no copy of F is created.

Technical work!

F = r=2

21

12

harmless

dangerous

This might be the same edge

“Bastard”

• Goal: force a copy of F .

• We fix the most dangerous ordering ¼ and force a copy of (the r-matched version of) F

that contains a copy of F.

• This is enforced if we get r edges all closing the central copy of a copy of .

• We do this inductively in e(F) rounds, doing a small subgraphs type variance calculation in each round.

• works out if .

Upper Bound Proof

4

43

4

2

34

12

344

43

4

F ¼

F

Conclusion

• The problem can be solved completely using the methods of first and second moment only.

• The notion of r-matched graphs allows a combinatorial interpretation of the threshold formula.

• Open questions: (ongoing work with Michael Krivelevich)

•What if r = r(n) 1?

•What if we want to create a copy of F as quickly as possible?

Thank you! Questions?