The Probabilistic Method in Graph Theory

Introduction Applications in Discrete Mathematics Some examples and results

The Probabilistic Method In Graph Theory

Ehssan Khanmohammadi

Department of Mathematics The Pennsylvania State University

February 25, 2010 Introduction Applications in Discrete Mathematics Some examples and results

What do we mean by the probabilistic method? Why use this method? The Probabilistic Method and Paul Erd˝os

The probabilistic method is a technique for proving the existence of an object with certain properties by showing that a random object chosen from an appropriate probability distribution has the desired properties with positive probability. Pioneered and championed by Paul Erd˝oswho applied it mainly to problems in combinatorics and number theory from 1947 onwards. Introduction Applications in Discrete Mathematics Some examples and results

What do we mean by the probabilistic method? Why use this method? An apocryphal story quoted from Molloy and Reed

At every combinatorics conference attended by Erd˝osin 1960s and 1970s, there was at least one talk which concluded with Erd˝os informing the speaker that almost every graph was a counterexample to his/her conjecture! Introduction Applications in Discrete Mathematics Some examples and results

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

Three facts about the probabilistic method which are worth bearing in mind: 1: Large and Unstructured Output Graphs The probabilistic method allows us to consider graphs which are both large and unstructured. The examples constructed using the probabilistic method routinely contain many, say 1010, nodes. Explicit constructions necessarily introduce some structuredness to the class of graphs built, which thus restricts the graphs considered. 3: Covers almost Every Graph Erd˝osdid not say some graph is a counterexample to your conjecture, but rather almost every graph is a counterexample to your conjecture.

Introduction Applications in Discrete Mathematics Some examples and results

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

2: Powerful and Easy to Use Erd˝oswould routinely perform the necessary calculations to disprove a conjecture in his head during a ﬁfteen-minute talk. Introduction Applications in Discrete Mathematics Some examples and results

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

2: Powerful and Easy to Use Erd˝oswould routinely perform the necessary calculations to disprove a conjecture in his head during a ﬁfteen-minute talk.

3: Covers almost Every Graph Erd˝osdid not say some graph is a counterexample to your conjecture, but rather almost every graph is a counterexample to your conjecture. Introduction Applications in Discrete Mathematics Some examples and results

Applications in Discrete Mathematics

One can classify the applications of probabilistic techniques in discrete 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? Surprisingly often it is much easier to prove this than it is to give an example of a structure that works.

Introduction Applications in Discrete Mathematics Some examples and results

2: Proof of the Existence of Certain Structures Choose a structure randomly (from a probability distribution that you are free to specify). Estimate the probability that it has the properties you want. Show that this probability is greater than 0, and therefore conclude that such a structure exists. Surprisingly often it is much easier to prove this than it is to give an example of a structure that works. The following result of Szele (1943) is ofttimes considered the ﬁrst use of the probabilistic method. Theorem (Szele 1943) There is a tournament T with n players and at least n!2−(n−1) Hamiltonian paths.

Introduction Applications in Discrete Mathematics Some examples and results

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 path which traces each node (exactly once) following the direction of the graph. Introduction Applications in Discrete Mathematics Some examples and results

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 path which traces each node (exactly once) following the direction of the graph.

The following result of Szele (1943) is ofttimes considered the ﬁrst use of the probabilistic method. Theorem (Szele 1943) There is a tournament T with n players and at least n!2−(n−1) Hamiltonian paths. Let X be the random variable counting the number of Hamiltonian paths. A permutation σ on the set of nodes deﬁnes a Hamiltonian path iff (σ(i), σ(i + 1)) is a directed edge for all 1 ≤ i < n.

Introduction Applications in Discrete Mathematics Some examples and results

Szele’s result (cont.)

Proof. Choose uniform distribution on all tournaments with n nodes. A permutation σ on the set of nodes deﬁnes a Hamiltonian path iff (σ(i), σ(i + 1)) is a directed edge for all 1 ≤ i < n.

Introduction Applications in Discrete Mathematics Some examples and results

Szele’s result (cont.)

Proof. Choose uniform distribution on all tournaments with n nodes. Let X be the random variable counting the number of Hamiltonian paths. Let Xσ be the indicator of “σ deﬁnes a Hamiltonian path”. −(n−1) P(Xσ = 1) = 2 . P P −(n−1) X = σ Xσ, thus E(X ) = σ E(Xσ) = n!2 . Conclusion: There is a tournament for which X is equal to at least E(X ).

Introduction Applications in Discrete Mathematics Some examples and results

Szele’s result (cont.)

Proof. Choose uniform distribution on all tournaments with n nodes. Let X be the random variable counting the number of Hamiltonian paths. A permutation σ on the set of nodes deﬁnes a Hamiltonian path iff (σ(i), σ(i + 1)) is a directed edge for all 1 ≤ i < n. −(n−1) P(Xσ = 1) = 2 . P P −(n−1) X = σ Xσ, thus E(X ) = σ E(Xσ) = n!2 . Conclusion: There is a tournament for which X is equal to at least E(X ).

Introduction Applications in Discrete Mathematics Some examples and results

Szele’s result (cont.)

Let Xσ be the indicator of “σ deﬁnes a Hamiltonian path”. P P −(n−1) X = σ Xσ, thus E(X ) = σ E(Xσ) = n!2 . Conclusion: There is a tournament for which X is equal to at least E(X ).

Introduction Applications in Discrete Mathematics Some examples and results

Szele’s result (cont.)

Let Xσ be the indicator of “σ deﬁnes a Hamiltonian path”. −(n−1) P(Xσ = 1) = 2 . Conclusion: There is a tournament for which X is equal to at least E(X ).

Introduction Applications in Discrete Mathematics Some examples and results

Szele’s result (cont.)

Let Xσ be the indicator of “σ deﬁnes a Hamiltonian path”. −(n−1) P(Xσ = 1) = 2 . P P −(n−1) X = σ Xσ, thus E(X ) = σ E(Xσ) = n!2 . Introduction Applications in Discrete Mathematics Some examples and results

Szele’s result (cont.)

Let Xσ be the indicator of “σ deﬁnes a Hamiltonian path”. −(n−1) P(Xσ = 1) = 2 . P P −(n−1) X = σ Xσ, thus E(X ) = σ E(Xσ) = n!2 . Conclusion: There is a tournament for which X is equal to at least E(X ). Remark 2 Szele conjectured that the maximum possible number of Hamiltonian paths in a tournament on n players is at most n! (2−o(1))n . Alon proved this conjecture in 1990 using the probabilistic method.

Introduction Applications in Discrete Mathematics Some examples and results

Remark 1 A player who wins all games would naturally be the tournament’s winner. However, there might not be such a player. A tournament for which every player loses at least one game is called a 1-paradoxical tournament. More generally, a tournament T = (V , E) is called k-paradoxical if for every k-element subset S of V there is a vertex v0 in V \ S such that v0 → v for each v ∈ S. By means of the probabilistic method Erd˝osshowed that, for any ﬁxed value of k, if |V | is suﬃciently large, then almost every tournament on V is k-paradoxical. Introduction Applications in Discrete Mathematics Some examples and results

Remark 2 Szele conjectured that the maximum possible number of Hamiltonian paths in a tournament on n players is at most n! (2−o(1))n . Alon proved this conjecture in 1990 using the probabilistic method. Ramsey (1929) showed that R(k, l) is ﬁnite for any two integers k, l. Theorem (Erd˝os(1947))

k n 1−(2) k/2 If k · 2 < 1, then R(k, k) > n. Thus R(k, k) > b2 c for each k ≥ 3.

Introduction Applications in Discrete Mathematics Some examples and results

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

Deﬁnition The Ramsey number R(k, l) is the smallest integer n such that in any 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 . Theorem (Erd˝os(1947))

k n 1−(2) k/2 If k · 2 < 1, then R(k, k) > n. Thus R(k, k) > b2 c for each k ≥ 3.

Introduction Applications in Discrete Mathematics Some examples and results

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

Ramsey (1929) showed that R(k, l) is ﬁnite for any two integers k, l. Introduction Applications in Discrete Mathematics Some examples and results

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

Ramsey (1929) showed that R(k, l) is ﬁnite for any two integers k, l. Theorem (Erd˝os(1947))

k n 1−(2) k/2 If k · 2 < 1, then R(k, k) > n. Thus R(k, k) > b2 c for each k ≥ 3. For any ﬁxed set R of k nodes, let XR be the indicator of being monochromatic for induced subgraph of R, and deﬁne X for the whole graph similarly. 1−(k) Clearly, P(XR = 1) = 2 2 , and by our assumption X n 1− k E(X ) = E(X ) = 2 (2) < 1 R k R

Introduction Applications in Discrete Mathematics Some examples and results

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

Proof.

Consider a random 2-coloring of Kn: Color each edge 1 1 independently with probability 2 of being red and 2 of being blue. 1−(k) Clearly, P(XR = 1) = 2 2 , and by our assumption X n 1− k E(X ) = E(X ) = 2 (2) < 1 R k R

Introduction Applications in Discrete Mathematics Some examples and results

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

Proof.

Consider a random 2-coloring of Kn: Color each edge 1 1 independently with probability 2 of being red and 2 of being blue.

For any ﬁxed set R of k nodes, let XR be the indicator of being monochromatic for induced subgraph of R, and deﬁne X for the whole graph similarly. Introduction Applications in Discrete Mathematics Some examples and results

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

Proof.

Consider a random 2-coloring of Kn: Color each edge 1 1 independently with probability 2 of being red and 2 of being blue.

For any ﬁxed set R of k nodes, let XR be the indicator of being monochromatic for induced subgraph of R, and deﬁne X for the whole graph similarly. 1−(k) Clearly, P(XR = 1) = 2 2 , and by our assumption X n 1− k E(X ) = E(X ) = 2 (2) < 1 R k R Note that if k ≥ 3 and we take n = b2k/2c, then

1+ k k n 1− k 2 2 n 2 (2) < · < 1, k k! 2k2/2

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

Introduction Applications in Discrete Mathematics Some examples and results

Proof continued. E(X ) < 1, thus, there exists a complete graph on n nodes with no monochromatic subgraph on k nodes, because the expected value, that is, the mean number of monochromatic subgraphs is less than one, where the mean is taken over all 2-colorings of Kn. So, R(k, k) > n. Introduction Applications in Discrete Mathematics Some examples and results

1+ k k n 1− k 2 2 n 2 (2) < · < 1, k k! 2k2/2

and hence R(k, k) > b2k/2c. Theorem (Caro (1979), Wei (1981)) α(G) ≥ P 1 . v∈V dv +1

Introduction Applications in Discrete Mathematics Some examples and results

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 of its elements are adjacent. The size of a maximal (with respect to inclusion) independent set in a graph G = (V , E) is denoted by α(G). Introduction Applications in Discrete Mathematics Some examples and results

Example 3: A Result of Caro and Wei

Theorem (Caro (1979), Wei (1981)) α(G) ≥ P 1 . v∈V dv +1 Let Xv be the indicator random variable for v ∈ I and P X = v∈V Xv = |I |. For each v, E(X ) = P(v ∈ I ) = 1 , since v ∈ I iff v is the v dv +1 least element among v and its neighbors. Hence E(X ) = P 1 , and so there exists a speciﬁc v∈V dv +1 ordering < with |I | ≥ P 1 . v∈V dv +1

Introduction Applications in Discrete Mathematics Some examples and results

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

I = { v ∈ V |{ v, w } ∈ E ⇒ v < w }. For each v, E(X ) = P(v ∈ I ) = 1 , since v ∈ I iff v is the v dv +1 least element among v and its neighbors. Hence E(X ) = P 1 , and so there exists a speciﬁc v∈V dv +1 ordering < with |I | ≥ P 1 . v∈V dv +1

Introduction Applications in Discrete Mathematics Some examples and results

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 and P X = v∈V Xv = |I |. Hence E(X ) = P 1 , and so there exists a speciﬁc v∈V dv +1 ordering < with |I | ≥ P 1 . v∈V dv +1

Introduction Applications in Discrete Mathematics Some examples and results

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 and P X = v∈V Xv = |I |. For each v, E(X ) = P(v ∈ I ) = 1 , since v ∈ I iff v is the v dv +1 least element among v and its neighbors. Introduction Applications in Discrete Mathematics Some examples and results

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 and P X = v∈V Xv = |I |. For each v, E(X ) = P(v ∈ I ) = 1 , since v ∈ I iff v is the v dv +1 least element among v and its neighbors. Hence E(X ) = P 1 , and so there exists a speciﬁc v∈V dv +1 ordering < with |I | ≥ P 1 . v∈V dv +1 Introduction Applications in Discrete Mathematics Some examples and results

Explicit Constructions and Algorithmic Aspects

The problem of finding a good explicit construction is often very difficult. Even the simple proof of Erd˝osthat there are red/blue colorings of graphs with b2k/2c nodes containing no monochromatic clique of size k leads to an open problem that seems very difficult. An Open Problem Can we explicitly construct a graph as described above with n ≥ (1 + )k nodes in time that is polynomial in n?

This problem is still wide open, despite considerable eﬀorts from many mathematicians. Introduction Applications in Discrete Mathematics Some examples and results

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.