Project work
Discrete Mathematics
Ram Gautam and Shaunna J. Weber
Math 230 Spring 2013: Graph Theory Historical Project
Submitted to Dr.Angela Kohlhaas
You will work with one other student on this project. It will be worth 70 points toward your final
grade.
The due dates are as follows: Part I on April 19 at 11:59pm, Part II on May 3 at 11:59pm.
The questions below are just those I came up with when attempting to read “Early Writings on
Graph Theory” critically. This is a bridge course. I am attempting to both expose you to new ideas
and connect ideas with which you are already familiar. The project explores the nature of proof, and
the questions below should help you reflect on that. You will need to do some research to answer
some of the questions below (especially the first few). Your answers should be in complete
sentences on a separate Word document. Use the equation editor to format any mathematical
formulas you use.
Part I (through paragraph 17)
a. What is topology; both according to the project and from your own research (cite your
sources)? Describe the field in your own words.
According to the project topology is defined as Leibniz’s ‘geometry of position’.
Basically, topology is the modern version of geometry, the study of all different sorts of
spaces. The thing that distinguishes different kinds of geometry from each other (including
topology here as a kind of geometry) is in the kinds of transformations that are allowed
before you really consider something changed. (This point of view was first suggested by
Felix Klein, a famous German mathematician of the late 1800 and early 1900's.)1
b. What fields of study arose from Leibniz’s “geometry of position”? List at least three
(only two are mentioned in the project).
The field of study that arose from Leibniz’s “geometry of position” is:
 Modern Graph theory
 Calculus to study curves
 Binary system of arithmetic2
Sub-fields
 Point set topology
 Algebraic and Geometric topology
 Computer science
1
http://www.math.wayne.edu/~rrb/topology.html
2
http://www-history.mcs.st-andrews.ac.uk/Biographies/Leibniz.html
c. Do you think a computer-assisted proof is a valid mathematical proof? Why or why
not? Is a proof done purely by hand somehow better? Why or why not?
I do not think computer-assisted proof is a valid mathematical proof because some
mathematicians believe that lengthy computer assisted proof are not real mathematical since
they involve so many logical steps that they are not practically verified by human beings. 3
There are some proof that has be assisted by computer such as four color theorem but still I
do not support the computer assisted because it is not true that the successfully executed
program gives the correct answer. There might be errors that computer ignores such a run
time error and redundancy in algorithms. When you use the number 5 for computer proof
you do not have to proof why 5 is an odd number. Hence, computer proofs are not valid for
all the cases. Computer cannot work on complex number. I am sure computer gives the value
of √𝑖 2 is 𝑖 which is in fact -1 in terms of complex number. Hence, it would be funny to
accept computer-assisted proof as a valid mathematical proof.
However, while doing hand written mathematical proofs we do not have to worry about
logical and syntax errors. Each and every steps of the proof are supported by valid reasoning
and facts. Only the complete work is accepted as the valid mathematical proof. When you
use even number numbers or odd number, you should clearly define why that is odd or even
number. Otherwise, the proof is not considered as the valid mathematical proof. Hence, hand
written proof is precise and easy to understand which is better than computer assisted proof.
d. Write a paragraph about Euler and his mathematical contributions, including
information NOT presented in the project (cite your sources).
Mathematical contributions of Euler those are not included in the project: 4
 Euler introduced much of the mathematical notation in use today, such as the notation
𝑓(𝑥) to describe a function and modern notation for the trigonometric functions.
 The use of the Greek letter π was also popularized by Euler.
 Euler was also credited for inventing complex number 𝑖 to denote √−1
 He also discovered Euler formula as for any real number as :
3
4
http://www.math.washington.edu/~billey/computer.proofs.html
http://en.wikipedia.org/wiki/Contributions_of_Leonhard_Euler_to_mathematics
e. What does Euler do in paragraph 2 upon hearing the Konigsberg Bridge problem, even
before solving the specific problem? Why do you think he did so?
After hearing the problem Euler started to formulate the general problem. He is attempting to
figure out the basic solution and establish the theory for the bridges crossing problem. He is
trying to give the solution of answer through acrid fact and valid reasoning.
Euler was a mathematician. Mathematicians always prove their answers with facts and
generalize the problem before reaching any conclusion. For example,
Why is 5 an odd integer but not an even integer?
2 x 0 +1=1
2 x 1+1=3
2 x 2+1=5
2 x 3+1=7
Hence, general formula for odd is 2 x some integer +1. Thus an integer a is called odd
provided there is an integer b such that 𝑎 = 2𝑏 + 1. Since, 5 meets the criteria of odd , hence
5 is an odd integer. This is how mathematician works.
Hence, Euler is also trying to figure out the formula for the possible bridge crossing problem
even before solving the specific problem. If the Konigsberg Bridge meets the given criteria
of Euler than we will be able make a journey in the bridges otherwise not.
f. What is a graph, mathematically speaking?
In mathematics, a graph is a representation of a set of objects where some pairs of the objects
are connected by links. The interconnected objects are represented by mathematical
abstractions are called vertices, and the links that connect some pairs are called edges.
Typically, a graph is depicted in diagrammatic form as a set of dots for the vertices, joined
by lines or curves for the edges. Graphs are one of the objects of study in discrete
mathematics.5
5
http://en.wikipedia.org/wiki/Graph_(mathematics)
g. List two questions that occurred to you while reading through this project. Then do some
research and answer them.
After reading this project the two questions that are in I am thinking in my head are given below:
6

What is the basic difference between Euler Path and Euler Circuit and how do we represent
them in Graph?
An Euler path is a path that passes through each edge of a graph exactly one time. An Euler
circuit is a circuit that passes through each edge of a graph exactly one time. The difference
between an Euler path and an Euler circuit is that an Euler circuit must start and end at the
same vertex.6

Proof the Hand Shaking Theorem from the graph.
Let us suppose, we are given the following graph:
The hand shaking theorem is generalized by the following formula:
https://docs.google.com/viewer?a=v&q=cache:ZzHwFp8NkCgJ:www.lhup.edu/maslam/LHUP%2520Fall%252008/MAT
H101/Chapter%252014/chapter14_2.ppt+difference+between+euler+path+and+euler+circuit+%2B+ppt&hl=en&gl=us
Now form the above graph the degree of the vertices of the edges is:
∑ 𝑑(𝑣) = 𝑑(1) + 𝑑(2) + 𝑑(3) + 𝑑(4) + 𝑑(5) + 𝑑(6) + 𝑑(7)
𝑣∈𝑉
= 2 + 2 + 3 + 1 + 1 + 1 + 0 = 10
𝑎𝑛𝑑 𝑤𝑒 ℎ𝑎𝑣𝑒 𝑜𝑛𝑙𝑦 5 𝑒𝑑𝑔𝑒𝑠
Hence, the hand shaking theorem is proved from the above graph.
h. Exercise 1. (How to read math tip: construct and write down your own example to play with
as you read, asking the same questions about your example that the text asks in general.)
The sketch of a graph with 5 vertices and 8 edges are given below:
1
5
4
3
2
The numbers 1, 2, 3, 4, and 5 represents the five vertices and the dark 8 back line pairs represent the
8 edges.
i. What do we call the type of question Euler decides to attempt to prove in 3?
After practically rejecting the strategy of solving the bridge-crossing problem Euler again
reformulates the problem in terms of sequence of letters (vertices) representing land masses and
hence therefore neglecting diagrams. This concept leads to the concept of Euler path and Euler
circuit.
Hence, Euler is trying to prove the problem of the seven bridges of Konigsberg in 3.
j. Exercise 2. (How to read math tip: carry the example as far as you can. Formulating a simple
concrete example will give you a better understanding of general abstract concepts.)
Note:





Island A represents landmass A
Island B represents landmass B
Island C represents landmass C
North Bank represents landmass N
South Bank represents landmass S
The land masses represent the number of vertices. Since, there are five different land masses namely
South Bank, North Bank, Island A, Island B, and Island C , we need to have five 5 capital letters to
represent an Euler path if we want to go all land masses.
However, if we want to go from Island C to Island A, then the Euler path can be simply CBA, where
C is the common land mass.
k. What do you notice about how Euler’s style of argument in 7-8 differs from that of modern
textbook proofs?
Euler had defined that in order to complete the Euler Path every edge of should be used exactly
once. However, it is not always necessary that we need to every edges/bridges to complete circuit.
Please refer the diagram below:
Euler theory also brought the concept that if the number of bridges is any odd number and if it is
increased by one, then the number of occurrences of A are half of the result regardless from where
we start. Lest look at the above figure for example:
Suppose we are crossing five bridges namely a, b, c, d, and e starting from A to B. the generals idea
to cross the bridge looks as:
𝑎→𝑏→𝑐→𝑑→𝑒
𝐴→𝐵→𝐴→𝐵→𝐴→𝐵
Hence, if we count the number of A or B to cross five bridges is 3. The basic difference that can
be found in Euler proofs is the use capitals letters to represent vertex and small letters to
represent the edges or bridges.
However the textbook, “A discrete Introduction” has used the number to represent the vertex. Let
G=(V,E) be a graph. A walk in G is a sequence of vertices with each vertex adjacent to the next: that
is,
𝑊 = (𝑣𝑜, 𝑣1 , … … … … … , 𝑣𝑙 ) with 𝑣𝑜 ~𝑣1 ~, … … … … … ~, 𝑣𝑙
The length of this walk is 𝑙. Note that we have started the subscript at Zero and that there are 𝑙 +
1 vertices on the walk.
Fig: Definition 48.1
For example, consider the graph in the figure.
 1~2~3~4 This is a walk of length three. It starts at vertex 1 and ends at vertex 4, and so we
call it a (1, 4) –walk.
Such kind of walk concept cannot be found in the Euler proofs.
l. Exercise 3. (How to read math tip: once you know the set-up for a problem, try to solve the
problem on your own before reading the proof. Not only will you learn the math more
thoroughly, but you might come up with an original proof!)
No for this problem the sequence does not seem to be possible.
From Paragraph five, Euler gave us an idea that the successive crossing of four bridges would be
represented by five letters, and in general, however many bridges the traveler crosses, his journey is
denoted by a number of letters one greater than the number of bridges. Thus crossing of seven
bridges requires eight letters to represent it.
Again in paragraph 8, Euler gave us idea that if the number of bridges is any odd number and if it is
increased by one, then the number of occurrences of landmasses (say A) are half of the result
regardless from where we start.
So from above diagram:
 Five bridges goes to landmass A , hence the number of occurrence of A is 3
 Three bridges goes to landmass B, hence the number of occurrence of B is 2
 Three bridges goes to land mass C, hence the number of occurrence of C is 2
 Three bridges goes to land mass D, hence the number of occurrence of D is 2
The total number vertices required are 9.
As the sequence of vertices is more than 8, such kind of journey is not possible.
m. Exercise 4. (How to read math tip: Don’t believe everything you read. The beauty of
mathematics is that you can check the claims for yourself.)
 Paragraph 9- I agree with paragraph 9 that the sequence of 9 vertices is not possible in seven
bridges of Konigsberg problem, hence the journey is not possible. Please refer the solution of
exercise 3 for this explanation.
 Paragraph 10- I do not agree with Euler that it is possible to tell whether a journey can be
made or not whenever the number of bridges leading to each area is odd, since we are not
still clear either journey can be made or not on Konigsberg bridge.
 Again Euler says that if the number of occurrences is greater than one more than the number
of bridges then, such journey can be never accomplished. From paragraph five, Euler says
that how many bridges the traveler crosses, his journey is denoted by a number of letter one
greater than the number of bridges. Hence, his logics are very confusing and hard to trust.
However, his analyses on the bridges are used in the modern text book proofs.
 Again in paragraph 13, Euler discussed that the required route can be found in the case
where total occurrences is equal the number of bridges plus one.
This can be states as, let the number of occurrences be O and e+1 be bridges. Then, if O=e+1
then, the route can be found.
Again using contradiction,
Let O be the number of occurrences and e be the number of bridges and O =e+1. Then for
the sake of contradiction, the route cannot be found. Using occurrence principle given by
Euler, In the case of the Konigsberg bridges, there must be three occurrences of the letter A
in the representation of the route, since five bridges (a, b, c, d, e) lead to the area A. Hence
the occurrences are:
𝑜𝑟, 3 ∗ 𝑂 = 5 − − − − − − − − − − − − − − 𝐼
5
𝑜𝑟, 𝑂 = − − − − − − − − − − − − − − − −𝐼𝐼
3
Now from equation I and I, it can be concluded that: O≠ 𝑒 + 1. Since, O is not an integer.
Thus, I am not convinced with Euler reasoning.
 The proofs given by are mostly thoroughly reasoning. He never used the proofs such as if
then else, if and only if, proof by contradiction and so on.
n. How is Euler’s argument in 14-15 both similar to and distinct from proof techniques you
have learned in class?
Both the paragraphs have the same concept of proof techniques that we have discussed in our class.
For our team, it looks like the modern proof techniques such as the proof by smallest counter
example and proof by induction had used by Euler in paragraph 14 and 15. But his proving
technique is much simple and does match the criteria of modern proofs. It also looks that the
paragraph 14 and 15 is the mixture these two proof techniques.
At first all the land masses are defined by capital letters and all the bridges are defined by small
letters. Since, bridges connect the land masses, the total number of bridges are increased by 1 which
is the similar concept of inductive step. Hence, the total numbers of bridges are calculated. This is
shows the similar property of proof by induction. The total number bridges are given to be 8 and 16
for paragraph 14 and 15 respectively.
Again, the second part is similar to smallest counter example. For this part total numbers of bridges
and land masses are tabulated. The number of bridges that reach to each land masses is added on the
basis of Euler principle. The number of bridges was found to be 9 and 16 in paragraph 14 and 15
respectively.
Hence, the either journey can be made or not is proofed by Euler in paragraph 14 and 15.
Unfortunately, they are no longer valid reasoning for the modern world.
o. Exercise 5.
Euler path: An Euler path is one that uses every edge of graph exactly once but it starts at one
vertex and ends at other vertex. Yes, the bridges crossing problem contains the Euler path.
p. How many distinct paths are there in your example? In particular, is the path unique? (This
is another type of question mathematicians pose.)
Distinct Euler Paths:
 Path 1:
 Path 2:
 Path 3:
 Path 4:
Since, there is more than one path. The path is not unique.
q. Reformulate paragraphs 16-17 using deg(v) notation. Do you think this adds clarity to
Euler’s argument? Why or why not?
In Paragraph 16, Euler argued that the number of bridges written next to the letters A, B , C, etc.
add up to twice the total number of bridges. This statement can be rewritten as the total number of
edges written next to vertices adds up to the twice the total number of edges. But the total number of
edges incident on vertex V is referred to as the degree of vertex V as deg(𝑣). Hence, it can be
concluded that the sum of deg(𝑣) is twice the number of edges. This statement is still ambiguous
because Euler had failed to define the meaning of “bridges written next to the letters”. Again, he had
also failed to clarify that this is a valid reasoning for finite graph too.
In Paragraph 17, Euler argued that the total number of bridges leading to each area must be an even
number, since half of it is equal to the number of bridges. Hence, if some of the number of bridges
attached to the letters A, B, C, etc. are odd, and then there must be an even number of these. This
statement can be restated as the total number of edges leading to vertices is even. Hence, the number
bridges attached to vertices are odd, and then there must be even vertices. Thus, it can be written as,
if deg(𝑣) is odd then there must even number of vertices. This is still ambiguous because it has
failed to define the where is this theory is valid or the type of graph it represents. In fact this is valid
reasoning for finite graph.
r. Exercise 6. (How to read math tip: compare what you read with other sources. If it’s a
famous result, it probably appears in a variety of formulations, some easier to read and
understand than others.)
A modern statement of the Handshake Theorem would be: The sum of the degree of all vertices in
a finite graph equals twice the number of edges in the graph.
According to the modern text book “A Discrete Introduction”, Let G= (V, E). The sum of the degree
of the vertices in G is twice the number of edges; that is,
∑ 𝑑(𝑣) = 2|𝐸|
𝑣∈𝑉
The modern text book has used combinatorial proof to prove the given theory, which means it has
given two answers and said that first answer is equal to the second answer. At first it will ask the
question as, “In how many ways…………….? At first it argues why first answer is correct to the
question. And secondly, it argues why second answer is correct. Since, first answer equal to the
second answer. Hence, the given proof is valid.
For example:
Suppose the vertex set is 𝑉 = {𝑣1 , 𝑣2 , … … … … . . , 𝑣𝑛 }. Now creating an n x n matrix of above graph
assuming that the entry in row 𝑖 and 𝑗 of above chart is 1 if 𝑣1 ~ 𝑣2 and is otherwise is 0. The matrix
looks like as:
0 1 1 0 0 0 0
1 0 1 0 0 0 0
1 1 0 1 0 0 0
0 0 1 0 0 0 0
0 0 0 0 0 1 0
0 0 0 0 1 0 0
[0 0 0 0 0 0 0]
How many 1’s are possible in this chart?
 Answer one: from above chart it can be said that for every G there are exactly two 1’s
possible. If 𝑣𝑖 𝑣𝑗 ∈ 𝐸, then there is a 1 in position 𝑖 𝑗 and a 1 in position 𝑗𝑖. Hence the
number of 1’s is exactly 2|𝐸|
 Answer two: let us consider the row corresponds to some vertex 𝑣𝑖 . There is a 1 in this row
exactly for those vertices adjacent to 𝑣𝑖 . Thus, the number of 1s in this row is exactly the
degree of the vertex- that is, d(𝑣𝑖 ). Hence the number of 1s in the chart is equals the sum of
the degree of the vertices of the graph.
Since, the both answers are correct to the question “How many 1s are in this chart?” Hence,
we can conclude that sum of the degree of the vertices of G (answer two) equals twice the
number of edges of G (answer one).
(From text book)
However, Euler had used observational method to verify his proof. With his observation he
came up with basic mathematics to proof his claim. Let the v be vertex and E be edge. So,
the number of edges written next to vertex adds twice the total number of bridges.
Since, the bridge joins to two areas the number of edges is E+E which equals 2E. Hence, the
given statement is true. This is very simple and incomplete reasoning while compared to
modern proof used in our text book. (Note: The symbols are given to show that the reasoning
of Euler was very week.)
But, the modern text book have also used same basic concept to proof their argument while
using adjacency matrix of the graph. The book also agrees that there are exactly two 1s in the
chart, which means two vertices are adjacent. In simple language two areas are connected by
bridges, the same concept of Euler.
s. Exercise 7.
Theorem: Every finite graph contains even vertices with odd degree.
The proof given by Euler is more ambiguous and unclear because, he had failed to define the
definition of odd and even. He had used the letters 3 and 5 as odd but had not defined why it is odd.
He had said if some of all numbers of bridges attached to the letters A, B, C, etc. are odd; there must
be an even number of these. Form the definition of degree of vertices, the above statement can be
easily said that if the degree of vertices is odd, then the graph contains even vertices. He had also not
given clear reasoning for the validity of this concept in graph. His proof was basically for the bridges
problems. However the logic used by Euler is still true and similar concept is used in modern proof
but his proof does not meet the criteria of modern proof techniques.
However, the modern text book has very different approach to proof the above theorem. Every little
term is defined very clearly that’s associated with the proof. It has defined the concept of self-loop
of the edge and complete graph, the reason for validity of this proof in graph, and so on. The text
book has given the concept of contrapositive to validate the above theorem. According to the text
book, “A discrete Introduction”, the proof by contrapositive can be proved by assuming not B and
working to prove not A for the statement if A then B.
The modern text book has also used the concept of induction to prove the above theorem. At first
the set of vertices is defined and basis step, n=1 or 2 is checked. Again the induction hypothesis is
given as (T-v) had (n-2) number of edges and from IH the given theorem is proved.
http://books.google.com/books?id=Be6t04pgggwC&pg=PA2&dq=every+finite+graph+contains+an
+even+number+of+vertices+with+odd+degree.&hl=en&ei=ILqvTZjjFNK_0QGl0rn1CA&sa=X&oi
=book_result&ct=result&resnum=6&ved=0CEYQ6AEwBQ#v=onepage&q&f=false
Part II
t. Exercise 8.
u. Exercise 9.
v. Exercise 10. Rewrite the proof in its entirety.
w. Exercise 11.
x. Exercise 12.
y. Which proof do you prefer? Why? Which is more rigorous? Which one is easier to follow? Why?
z. Based on this project and your class experience, how has proof-writing changed over the last few
centuries? What is mathematical truth and how do you justify it?
aa. Dictionary: Construct a list of graph theory terms introduced in the project. Your dictionary
should include a definition and example for each term. You will be expected to know and
understand these concepts after handing in your project.