Small Subgraphs in Random Graphs and the Power of Multiple Choices

The Online Case

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

Introduction

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

• Achlioptas process (named after Dimitris Achlioptas):

• 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 theappearance of Fbe avoided havingthis freedom of

choice?

F = , r=2

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

N0=N0(F, r, n)

N = N(n) = numberof steps in the

Achlioptas process

There is a strategy that avoids

creating a copy of F withprobability 1-o(1)(as n tends to infinity)

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

• Krivelevich et al. conjecture a general threshold formula

• We disprove this conjecture and solve the problem in full generality

... we will develop an intuition for the

threshold formula in the following

r-matched graphr=2

Graph

r-matched Graphs

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

- randomly partition the m edges into sets of size r

Achlioptas process after N stepsis distributed as

Threshold for the appearanceof small subgraphs in Gn,m (Bollobás, 1981):

m=m(n)

The Gluing IntuitionAnalogue of Bollobás‘ theoremfor r-matched graphs:

m=m(n)

Our key idea:relate Achlioptas processto ‘static’ objectF = , r=2

Greedy strategy:

e/v = 5/4

As long asthis subgraph does not appear,hence we do not lose.

“GluingIntuition”

J

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 this subgraph does not

appear, 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

Optimal Strategy for ¼1:

Edge ordering ¼1:

1

12

2

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

The F-Avoidance-Strategy (1)Measure the harmlessness of asubgraph by the parameter

21

harm

les

sdangero

us

F = , r=2

The F-Avoidance-Strategy (2)

• 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

• Our strategy considers ordered subgraphs

Lower Bound ProofLemma: By our strategy, each black copy of some ordered graph is contained in a copy of some rare grey-black r-matched graph H’.

1

2

H’“History graph”

F does not appear

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

• Below the threshold a.a.s. noneof the histories H’ appears in

Technical work!

F = r=2

21

12

harmless

dangerous

This might be the same edge

“Bastard”

rare: expectation at mostimplies that H’ does not appear in with probability 1-o(1)

• Goal: force a copy of F, in some fixed order regardless of the strategy

• Multiround approach (#rounds = #edges of F)

• In each round: count how many copies evolvefurther, 1st+2nd MM, small subgraphvariance calculation for r-matched graphs

• Optimize for the best upper bound

Upper Bound Proof

F = , r=2

1st round

2nd round

3rd round

Thank you! Questions?