§5.6 Problems: extremal graph theory
There are many good problems in the book. I recommend for this set of notes you look primarily at the exercises from Section 7.3 and at the ones from Sections 11.1 and 11.2 for practice problems. Below, you have some additional extremal graph theory problems.
Give them a try if you have time! Some of them will be discussed by your TA during the upcoming discussion sessions from 4 to 5 PM on Tuesdays and Thursdays.
Problem 5.6.1. Prove that a graph with n vertices and k edges has at least k
3n(4k − n2) triangles.
Problem 5.6.2. Prove Zarankiewicz’s Lemma: In any graph G with no Kr subgraph, prove that there exists a vertex with degree at most br−2r−1nc.
Problem 5.6.3. Use Zarankiewicz’s Lemma to give another proof of Turan’s theorem.
Problem 5.6.4. For a pair A = (x1, y1) and B = (x2, y2) of points in the plane, let d(A, B) = |x1− x2| + |y1− y2|.
We call a pair (A, B) of (unordered) points harmonic if 1 < d(A, B) ≤ 2. Determine the maximum number of harmonic pairs among 100 points in the plane.
Problem 5.6.5. A graph G has n2+ 1 edges and 2n vertices.
a) Prove that it contains two triangles sharing a common edge.
b) Prove that it contains at least n triangles.
Problem 5.6.6. A graph G has n vertices and no triangles. No matter how we partition its vertices into two classes, there are two adjacent vertices in the same class. Prove that some vertex has degree at most 2n/5.
Problem 5.6.7. The edges of the complete graph on 2n+ 1 vertices are colored in one of n colors. Prove that there is a monochromatic cycle of odd length.
Problem 5.6.8. We have n charged batteries, n uncharged batteries and a radio which needs two charged batteries to work. Suppose we don’t know which batteries are charged and which ones are uncharged. Find the least number of attempts sufficient to make sure the radio will work. An attempt consists in putting two batteries in the radio and check if the radio works or not.
Problem 5.6.9. Given a graph with 2n + 1 vertices, such that for any n vertices, there exists another one connected to all these n vertices. Show that there is a vertex connected to all the other 2n vertices.
Problem 5.6.10. A graph G has n vertices and nk/2 edges, with k ≥ 1. Prove that the maximum number of pairwise nonadjacent vertices of G is at least k+1n .
Problem 5.6.11. A graph G with 2n vertices is given. For every n different vertices of G, there exists a vertex connected with all of them. What is the minimal possible number of edges of G ?
Problem 5.6.12. The vertices of a finite connected graph cannot be colored with less than n + 1 colors so that adjacent vertices have different colors. Prove that n(n − 1) edges can be removed from the graph so that it remains connected. 2
6 Probabilistic Method
Once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth.
—Arthur Conan Doyle This is one of the most powerful tools in modern combinatorics. It was used at first by our recurring hero, Erd˝os, who proved a great deal of impressive results using it. But perhaps more importantly, he introduced a completely new perspective in combinatorics:
one can use randomness (i.e. tools from probability theory) to prove results about concrete objects, such as sets or real numbers. In our last few classes in Math 244, we will scratch the surface of this theory by discussing a few important examples, which will illustrate the main idea. The canonical reference is the wonderful book by Noga Alon and Joel Spencer called The Probabilistic Method.
§6.1 The basic tools
We will only work with finite probability spaces. Let me collect the following basic definitions:
Definition 6.1.1. A (finite) probability space is the data of a finite set Ω (called the sample space) and of a map (called the probability distribution) P : Ω → [0, 1] such that
X
ω∈Ω
P (ω) = 1.
The subsets A of Ω are called events. If A is an event, its probability is defined as P (A) =X
x∈A
P (x).
A random variable X is simply a map X : Ω → R. The expectation (or mean value) of X is
E[X] = X
ω∈Ω
X(ω)P (ω) =X
x∈R
x · P (X = x).
Note that the second sum is finite, as the image of X is finite.
An important example of probability distribution is the uniform distribution, namely the constant map |Ω|1 , for which P (A) = |A||Ω| for all events A.
Proposition 6.1.2
For any events A, B we have P (A∪B)+P (A∩B) = P (A)+P (B) and the probability of the complement of A is 1 − P (A).
Proof. This is immediate from the definition.
6.1 The basic tools
Definition 6.1.3. a) Events A1, A2, ..., Akare called independent if for all I ⊂ {1, 2, ..., k}
we have
P (∩i∈IAi) =Y
i∈I
P (Ai).
b) Two random variables X, Y are called independent if the events X = a and Y = b are independent for all a, b.
Proposition 6.1.4
If X, Y are independent random variables, then E[XY ] = E[X]E[Y ].
The proof is an easy exercise left to the reader. This proposition is very useful!
Here’s the basic idea of the probabilistic method: if you want to show the existence of an object with properties P1, ..., Pk, it is enough to prove that there is a probability space (Ω, P ) such that the sum of the probabilities that an object does not have property Pi is smaller than 1.
Theorem 6.1.5
Let (Ω, P ) be a finite probability space. For any subsets A1, A2, ..., Ak of Ω we have
P (∪ki=1Ai) ≤
k
X
i=1
P (Ai),
with equality if the events are pairwise disjoint. In particular, if Pk
i=1P (Ai) < 1, then ∪Ai6= Ω.
Proof. Let B1 = A1 and let Bi be the complement of ∪i−1j=1Aj in Ai for all i ≥ 2. Then clearly ∪jBj = ∪jAj and the Bj’s are pairwise disjoint. But then it is clear that P (Bj) ≤ P (Aj) (as Bj ⊂ Aj and P takes nonnegative values) and
P (∪jAj) = P (∪jBj) =X
P (Bj) ≤X
P (Aj).
The rest is immediate.
Theorem 6.1.6 (Linearity of Expectation) If X1, X2, ..., Xk are random variables, then
E[X1+ X2+ ... + Xk] = E[X1] + E[X2] + ... + E[Xk].
Proof. This is an immediate consequence of the definition of expectation.
When using the probabilistic method, one is naturally confronted with estimating probabilities, which sometimes can be quite painful. The following two inequalities are very basic, but useful tools in estimating probabilities: