Download AN EMORY MATH CIRCLE: INTRODUCTION TO GRAPH THEORY

AN EMORY MATH CIRCLE: INTRODUCTION TO GRAPH THEORY
JACKSON S. MORROW
1. W HAT IS A G RAPH ?
Graphs can represent a lot of different things such as road maps, electrical networks,
and even schedules! A graph is a representation of a set of points and how they behave
together. To get a better understanding, we need some formal definitions.
Definition 1.1. A graph consists of vertices and edges. You can think of vertices as points,
and edges are lines that connect some pairs of points.
Definition 1.2. A pair of vertices which are connected by an edge are called adjacent.
Definition 1.3. The degree of a vertex is the number of edges going out of it.
We will only be considering simple graphs, which are graphs with no self-loops, and
no multiple edges between two vertices. Also, we will only be considering undirected
graphs, which means that there is no distinction between the two vertices connected by
an edge. We now gives some definitions of some graphs that we will be considering today
(I have left some space between the definitions so you can draw a picture of what these
graphs look like).
Definition 1.4. A tree on n vertices Tn is an undirected graph in which any two vertices
are connected by exactly one path.
Definition 1.5. A path on n vertices Pn is a specific kind of tree, namely a tree that contains
only vertices of degree 2 or 1.
Date: November 14, 2015.
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Definition 1.6. A cycle graph on n vertices Cn is a graph that consists of a single cycle or
in other words some number of vertices connected in a chain.
Definition 1.7. A complete graph on n vertices Kn is a simple, undirected graph in which
every pair of distinct vertices is connected by a unique edge.
Although most of the graphs we consider are all nice and symmetric, we do not care
how a graph is drawn-the only thing that matters is whether there exists an edge between
a pair of vertices or not. An edge can be a straight line, a curve, or anything you want, as
long as it connects the two vertices.
Exercise 1.8. In this exercise, you will count the number of edges of the above graph
types.
(1) Take a path with 5 vertices. How many edges does it have? How about a path
with 6 vertices? n vertices?
(2) Take a cycle graph with 5 vertices. How many edges does it have? How about a
cycle graph with 6 vertices? n vertices?
(3) Take a complete graph with 4 vertices. How many edges does it have? How about
a complete with 5 vertices? 6 vertices? n vertices?
Exercise 1.9. In this exercise, you will count the degree of the vertices of the above graph
types.
(1) Take a path with 5 vertices. What are the degrees of the vertices? How about a
path with 6 vertices? n vertices?
(2) Take a cycle graph with 5 vertices. What are the degrees of the vertices? How
about a cycle graph with 6 vertices? n vertices?
(3) Take a complete graph with 5 vertices. What are the degrees of the vertices? How
about a complete graph with 6 vertices? n vertices?
Exercise 1.10. In this exercise, you will count the number of graphs on a specific number
of vertices.
(1) How many graphs are there with exactly 2 vertices? Draw all of them!
(2) How many graphs are there with exactly 3 vertices? Draw all of them!
(3) How many graphs are there with exactly 4 vertices? Draw all of them! (Hint: there
are 11).
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2. C OLORING
Definition 2.1. A vertex coloring of a graph is an assignment of colors to each vertex of
the graph in such a way that no two adjacent vertices have the same color. An interesting
question in general is to find the minimum number of colors needed to color a graph.
This minimum number is called the chromatic number of the graph.
Exercise 2.2. In this exercise, you will compute the chromatic number of the above graph
types.
(1) Take a path with 5 vertices. How many colors do you need to color it? How about
a path with 6 vertices? n vertices?
(2) Take a cycle graph with 4 vertices. How many colors do you need to color it? How
about a cycle graph with 5 vertices? 6 vertices? 7 vertices? n vertices?
(3) Take a complete graph with 4 vertices. How many colors do you need to color it?
How about a complete graph with 5 vertices? 6 vertices? n vertices?
3. R AMSEY T HEORY
Before we can discuss Ramsey theory, we need to recall the Pigeonhole Principle.
Pigeonhole Principle. If I have 3 pigeons and 2 pigeonholes, then some hole must have
at least 2 pigeons in it. More generally, if I have n + 1 pigeons and n pigeonholes, then
some hold must have at least 2 pigeons in it.
Here is the type of question that we can answer using the Pigeonhole Principle and
colorings.
Question. Show that in any group of 6 people that there are 3 mutual acquaintances or 3
mutual non-acquaintances.
Proof. We assume that there is a relationship (either acquaintances or non-acquaintances)
between each of the 6 people. Let’s label our group of 6 people as tA, B, C, D, E, Xu. Pick
one person out the the group of 6, we shall denote this person by X. We know that X
is either a mutual acquaintance or a mutual non-acquaintance to tA, B, C, D, Eu. We can
think of this statement as follows: X is a degree 5 vertex and each edge is colored red or
blue.
The Pigeonhole Principle says that at least three of them must be the same color; let’s
say that they are colored red. Let A, B, C be the adjacent vertices of the three edges that
are colored red. Now if one of the edges AB, BC, CA is red, then that edge together with
the other two edges from X forms a red triangle. If none of AB, BC, CA is red, then all
three edges are blue and we have a blue triangle, namely ABC.
We just did the first (non-trivial) example of a Ramsey theory problem! We actually
show that if we color the edges a complete graph on 6 vertices with red and blue, then we
either get a red triangle or blue triangle.
Definition 3.1. Let r (Km , Kn ) be the smallest integer p such that if we two color the edges
of K p with red and blue, we either get a red Km or a blue Kn .
Example 3.1.1. Above we showed that r (K3 , K3 ) ď 6. To prove that r (K3 , K3 ) = 6, we need
to find a coloring of K5 with red and blue that does not contain a triangle with all edges
the same color. Can you think of one?
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4. E XTRA E XERCISES
Exercise 4.1. The below graph is called the Petersen graph:
How many colors do you need to color the Petersen graph? (Hint: First find a lower
bound on the chromatic number by noticing a cycle graph within the Petersen graph.)
Exercise 4.2. A student has 37 days to prepare for an exam. She knows she will require
no more than 60 hours of study. She wishes to study at least 1 hour per day. Show that no
matter how she schedules her study time (in an integer number of hours per day), there
is a succession of days during which she will have studied exactly 13 hours.
ř
řday 2
Proof. Let a1 := # of hours in day 1, a2 = day 1 # hours, and a37 = # of hours each day
for 37 days. Since she wishes to study at least 1 hour per day, we have that a1 ă a2 , ¨ ¨ ¨ , a37 .
We also know that a37 ď 60.
Now, here is the trick! Consider the sequence a1 + 13, a2 + 13 . . . , a37 + 13, we shall
denote the elements of this sequence by bi . Again this is strictly increasing and a37 + 13 ď
73. Note that we have 2 ¨ 37 = 74 numbers and each is between 1 and 73, so the Pigeonhole
Principle apples i.e., two of these numbers must agree.
Since our sequences are strictly increasing, we have that ai ‰ a j and bi ‰ b j , which
implies that ai = b j for some i and j. Moreover, we have that ai = a j + 13 for some
i ‰ j. Moreover, the days j + 1, j + 2, . . . , i are the days in which she studied for exactly 13
hours.
Exercise 4.3. Use the Pigeonhole Principle to answer the following question. Say a chessmaster has 11 weeks to prepare for a tournament. He wants to play a match everyday,
but does not want to get burned out i.e., no more than 12 matches in 1 week. Show that
there is a succession of days when he plays exactly 21 matches.
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