In Graph Theory Ehssan Khanmohammadi

Introduction Applications in Discrete Mathematics Some examples and results

The Probabilistic MethodIn Graph Theory

Ehssan Khanmohammadi

Department of MathematicsThe Pennsylvania State University

February 25, 2010

What do we mean by the probabilistic method? Why use this method?

The Probabilistic Method and Paul Erdos

The probabilistic method is a technique for proving theexistence of an object with certain properties by showing thata random object chosen from an appropriate probabilitydistribution has the desired properties with positive probability.

Pioneered and championed by Paul Erdos who applied itmainly to problems in combinatorics and number theory from1947 onwards.

An apocryphal story quoted from Molloy and Reed

At every combinatorics conference attended by Erdos in 1960s and1970s, there was at least one talk which concluded with Erdosinforming the speaker that almost every graph was acounterexample to his/her conjecture!

Three facts about the probabilistic method which are worthbearing in mind:

1: Large and Unstructured Output Graphs

The probabilistic method allows us to consider graphs which areboth large and unstructured.

The examples constructed using the probabilistic methodroutinely contain many, say 1010, nodes.

Explicit constructions necessarily introduce somestructuredness to the class of graphs built, which thusrestricts the graphs considered.

2: Powerful and Easy to Use

Erdos would routinely perform the necessary calculations todisprove a conjecture in his head during a fifteen-minute talk.

3: Covers almost Every Graph

Erdos did not say some graph is a counterexample to yourconjecture, but rather almost every graph is a counterexample toyour conjecture.

2: Powerful and Easy to Use

Erdos would routinely perform the necessary calculations todisprove a conjecture in his head during a fifteen-minute talk.

3: Covers almost Every Graph

Erdos did not say some graph is a counterexample to yourconjecture, but rather almost every graph is a counterexample toyour conjecture.

Applications in Discrete Mathematics

One can classify the applications of probabilistic techniques indiscrete mathematics into two groups.

1: Study of Random Objects (Graphs, Matrices, etc.)

A typical problem is the following: if we pick a graph “at random,”what is the probability that it contains a Hamiltonian cycle?

2: Proof of the Existence of Certain Structures

Choose a structure randomly (from a probability distributionthat you are free to specify).

Estimate the probability that it has the properties you want.

Show that this probability is greater than 0, and thereforeconclude that such a structure exists.

Surprisingly often it is much easier to prove this than it is to givean example of a structure that works.

2: Proof of the Existence of Certain Structures

Choose a structure randomly (from a probability distributionthat you are free to specify).

Estimate the probability that it has the properties you want.

Show that this probability is greater than 0, and thereforeconclude that such a structure exists.

Surprisingly often it is much easier to prove this than it is to givean example of a structure that works.

Example 1: Szele’s result

Two Definitions

A directed complete graph is called a tournament.

By a Hamiltonian path in a tournament we mean a pathwhich traces each node (exactly once) following the directionof the graph.

The following result of Szele (1943) is ofttimes considered the firstuse of the probabilistic method.

Theorem (Szele 1943)

There is a tournament T with n players and at least n!2−(n−1)

Hamiltonian paths.

Example 1: Szele’s result

Two Definitions

A directed complete graph is called a tournament.

By a Hamiltonian path in a tournament we mean a pathwhich traces each node (exactly once) following the directionof the graph.

The following result of Szele (1943) is ofttimes considered the firstuse of the probabilistic method.

Theorem (Szele 1943)

There is a tournament T with n players and at least n!2−(n−1)

Hamiltonian paths.

Szele’s result (cont.)

Proof.

Choose uniform distribution on all tournaments with n nodes.

Let X be the random variable counting the number ofHamiltonian paths.

A permutation σ on the set of nodes defines a Hamiltonianpath iff (σ(i), σ(i + 1)) is a directed edge for all 1 ≤ i < n.

Let Xσ be the indicator of “σ defines a Hamiltonian path”.

P(Xσ = 1) = 2−(n−1).

X =∑

σ Xσ, thus E (X ) =∑

σ E (Xσ) = n!2−(n−1).

Conclusion: There is a tournament for which X is equal to atleast E (X ).

Proof.

P(Xσ = 1) = 2−(n−1).

X =∑

σ E (Xσ) = n!2−(n−1).

Proof.

P(Xσ = 1) = 2−(n−1).

X =∑

σ E (Xσ) = n!2−(n−1).

Proof.

P(Xσ = 1) = 2−(n−1).

X =∑

σ E (Xσ) = n!2−(n−1).

Proof.

P(Xσ = 1) = 2−(n−1).

X =∑

σ E (Xσ) = n!2−(n−1).

Proof.

P(Xσ = 1) = 2−(n−1).

X =∑

σ E (Xσ) = n!2−(n−1).

Proof.

P(Xσ = 1) = 2−(n−1).

X =∑

σ E (Xσ) = n!2−(n−1).

Remark 1

A player who wins all games would naturally be the tournament’swinner. However, there might not be such a player. A tournamentfor which every player loses at least one game is called a1-paradoxical tournament. More generally, a tournamentT = (V ,E ) is called k-paradoxical if for every k-element subset Sof V there is a vertex v0 in V \ S such that v0 → v for eachv ∈ S . By means of the probabilistic method Erdos showed that,for any fixed value of k, if |V | is sufficiently large, then almostevery tournament on V is k-paradoxical.

Remark 2

Szele conjectured that the maximum possible number ofHamiltonian paths in a tournament on n players is at most

n!(2−o(1))n . Alon proved this conjecture in 1990 using theprobabilistic method.

Remark 1

A player who wins all games would naturally be the tournament’swinner. However, there might not be such a player. A tournamentfor which every player loses at least one game is called a1-paradoxical tournament. More generally, a tournamentT = (V ,E ) is called k-paradoxical if for every k-element subset Sof V there is a vertex v0 in V \ S such that v0 → v for eachv ∈ S . By means of the probabilistic method Erdos showed that,for any fixed value of k, if |V | is sufficiently large, then almostevery tournament on V is k-paradoxical.

Remark 2

Szele conjectured that the maximum possible number ofHamiltonian paths in a tournament on n players is at most

n!(2−o(1))n . Alon proved this conjecture in 1990 using theprobabilistic method.

Example 2: Lower bound for diagonal Ramsey numbersR(k , k)

Definition

The Ramsey number R(k , l) is the smallest integer n such that inany two-coloring of the edges of a complete graph on n vertices Kn

by red and blue, either there is a red Kk or there is a blue Kl .

Ramsey (1929) showed that R(k , l) is finite for any two integersk , l .

Theorem (Erdos (1947))

)· 21−(k2) < 1, then R(k, k) > n. Thus R(k , k) > b2k/2c for

each k ≥ 3.

Definition

each k ≥ 3.

Definition

each k ≥ 3.

Lower bound for R(k , k) (cont.)

Proof.

Consider a random 2-coloring of Kn: Color each edgeindependently with probability 1

2 of being red and 12 of being

For any fixed set R of k nodes, let XR be the indicator ofbeing monochromatic for induced subgraph of R, and defineX for the whole graph similarly.

Clearly, P(XR = 1) = 21−(k2), and by our assumption

E (X ) =∑R

E (XR) =

)21−(k2) < 1

Proof.

E (X ) =∑R

E (XR) =

)21−(k2) < 1

Proof.

E (X ) =∑R

E (XR) =

)21−(k2) < 1

Proof continued.

E (X ) < 1, thus, there exists a complete graph on n nodes with nomonochromatic subgraph on k nodes, because the expected value,that is, the mean number of monochromatic subgraphs is less thanone, where the mean is taken over all 2-colorings of Kn. So,R(k , k) > n.

Note that if k ≥ 3 and we take n = b2k/2c, then(n

)21−(k2) <

k!· nk

2k2/2< 1,

and hence R(k , k) > b2k/2c.

Proof continued.

E (X ) < 1, thus, there exists a complete graph on n nodes with nomonochromatic subgraph on k nodes, because the expected value,that is, the mean number of monochromatic subgraphs is less thanone, where the mean is taken over all 2-colorings of Kn. So,R(k , k) > n.Note that if k ≥ 3 and we take n = b2k/2c, then(

)21−(k2) <

k!· nk

2k2/2< 1,

and hence R(k , k) > b2k/2c.

Example 3: A Result of Caro and Wei

A Definition and a Notation

A subset of the nodes of a graph is called independent if no two ofits elements are adjacent. The size of a maximal (with respect toinclusion) independent set in a graph G = (V ,E ) is denoted byα(G ).

Theorem (Caro (1979), Wei (1981))

α(G ) ≥∑

v∈V1

dv+1 .

Example 3: A Result of Caro and Wei

A Definition and a Notation

A subset of the nodes of a graph is called independent if no two ofits elements are adjacent. The size of a maximal (with respect toinclusion) independent set in a graph G = (V ,E ) is denoted byα(G ).

Theorem (Caro (1979), Wei (1981))

α(G ) ≥∑

v∈V1

dv+1 .

Proof.

Let < be a uniformly chosen total ordering of V . Define

I = { v ∈ V |{ v ,w } ∈ E ⇒ v < w }.

Let Xv be the indicator random variable for v ∈ I andX =

∑v∈V Xv = |I |.

For each v , E (Xv ) = P(v ∈ I ) = 1dv+1 , since v ∈ I iff v is the

least element among v and its neighbors.

Hence E (X ) =∑

v∈V1

dv+1 , and so there exists a specific

ordering < with |I | ≥∑

v∈V1

dv+1 .

Proof.

I = { v ∈ V |{ v ,w } ∈ E ⇒ v < w }.

∑v∈V Xv = |I |.

Hence E (X ) =∑

v∈V1

dv+1 .

Proof.

I = { v ∈ V |{ v ,w } ∈ E ⇒ v < w }.

∑v∈V Xv = |I |.

Hence E (X ) =∑

v∈V1

dv+1 .

Proof.

I = { v ∈ V |{ v ,w } ∈ E ⇒ v < w }.

∑v∈V Xv = |I |.

Hence E (X ) =∑

v∈V1

dv+1 .

Explicit Constructions and Algorithmic Aspects

The problem of finding a good explicit construction is often verydifficult. Even the simple proof of Erdos that there are red/bluecolorings of graphs with b2k/2c nodes containing nomonochromatic clique of size k leads to an open problem thatseems very difficult.

An Open Problem

Can we explicitly construct a graph as described above withn ≥ (1 + ε)k nodes in time that is polynomial in n?

This problem is still wide open, despite considerable efforts frommany mathematicians.

Thank You!References:

1 Alon, Spencer, The Probabilistic Method.

2 Gowers, et al., The Princeton Companion to Mathematics.

3 Molloy, Reed, Graph Colouring and the Probabilistic Method.

In Graph Theory Ehssan Khanmohammadi

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