Graph Theory Review: Sections 46,47,48,49,51
Section 46: Fundamentals
Denitions:
Graph: A graph is a pair G = (V, E ) where V is a nite set and E is a set of two-element subsets
of V .
Adjacent: Suppose G = (V, E ) is a graph and u, v V . Then, u is adjac

Discrete Mathematics (550.171)
Homework 9 (Due Friday, April 26, 2013)
Objectives: The student will gain experience with
The Chinese Remainder Theorem
Prime factorization
Finding modular square roots
General Directions: You must show all work and docum

Assignment #11 Solutions
Page 331, 38.2 Factor the following positive integers into primes.
a. 25 = 52 .
b. 4200 = 23 3 52 7.
c. 1010 = 210 510 .
d. 19 = 19.
e. 1 = 1.
Page 331, 38.4 Suppose a is a positive integer and p is a prime. Prove that p a if and

DISCRETE MATH HW 1 SOLUTIONS
1. Scheinerman 3.2: Here is a possible alternative to Denition 3.2: We say
that a is divisible by b provided a is an integer. Explain why this alternative
b
denition is dierent from Denition 3.2. Here dierent means that Deniti

An Introduction to Graph Theory
Instructors: Marisa and Paddy
Lecture 3: Trees and Art Galleries
Week 1 of 1
1
Mathcamp 2010
Glossary
cut-edge A cut-edge in a graph G is an edge whose removal increases the number of
connected components of G.
cut-vertex A

Home Exam (example)
Remarks: All the graphs here are without self loops and parallel or anti-parallel edges.
In all the algorithms, always explain their correctness and analyze their complexity. The
complexity should be as small as possible. A correct alg

[ADN.cn][Library]Homework
Mathematics for Computer Science: Homework 7
Botao Hu (Amber)
Mathematics for Computer Science: Homework 7
Instructed by Andrew C. Yao
Due on April 17, 2008
Botao Hu J72 2007011292
[email protected]
Contents
1 Exercise 7.3.5
2
2

CS 70 Fall 2004
P RINT your name:
Discrete Mathematics for CS Rao
,
(last)
Midterm 1
(rst)
S IGN your name: P RINT your username on cory.eecs: W RITE your section number (101 or 102):
Three sides of 8.5 by 11 sheet of notes is permitted. No calculators ar

Number theory lectures
By Dr. Mohammed M. AL-Ashker
Associated professor
Mathematics Department
E.mail:mashker @ mail.iugaza.edu
Islamic University of Gaza P.O.Box 108, Gaza, Palestine
1
Contents
1 Divisibility Theory of integers
1.1 The Division Algorith

Section 3
Prime Numbers and Prime
Factorisation
Denition 3.1 A prime number is an integer p > 1 whose only positive
divisors are 1 and p.
Example 3.2 The rst few prime numbers are:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, . . . .
See the Prime Page

CS 173, Spring 2009 Homework 10 Solutions (Total point value: 50 points.)
1. [10 points] Paths and Circuits in Graphs (a) Under what conditions does the graph Km,n have an Eulerian circuit? What has to be true about m and n? [Solution] m and n must both b

2
Walks, trails, paths, circuits and cycles
Denition 2.1. Let x and y be vertices in a graph G. An xy -walk is a nite
alternating sequence of vertices and edges:
x = x0 , e1 , x1 , e2 , e2 , . . . , xn1 , en , xn = y
where ei = xi1 xi for i = 1, . . . n.

Discrete Mathematics
Lent 2009
MA210
Solutions to Exercises 6
(1) Let V = cfw_1, 2, . . . , n. How many different graphs with vertex set V are there?
Solution. Each graph G with vertex set V is uniquely determined by its edge set E . E
must be a subset of

CS325 Midterm, Summer 2009, OSU
Forrest Briggs
July 29, 2009
1
Rules
You are required to type your solution, and turn it in printed at the beginning of class, Thursday,
7/23.
You are forbidden to discuss this test in any way with any person other than the

Discrete Mathematics (550.171)
Homework 10 (Due Friday, May 03, 2013)
Objectives: The student will gain experience with graphs.
General Directions: You must show all work and document any assumptions to receive
full credit. All problems are to be done by

GRAPH THEORY LECTURE 2
STRUCTURE AND REPRESENTATION PART A
Abstract. Chapter 2 focuses on the question of when two graphs are to be regarded as the same, on
symmetries, and on subgraphs. 2.1 discusses the concept of graph isomorphism. 2.2 presents symmetr

Chapter 10
Eulerian and Hamiltonian Paths
Circuits
This chapter presents two well-known problems. Each of them asks for a special kind
of path in a graph. If there exists such a path we would also like an algorithm to nd it.
Both of the types of paths (Eu

Discrete Mathematics (550.171)
Homework 8 (Due Friday, April 19, 2013)
Objectives: The student will gain experience with
The Division Theorem
Euclids GCD Algorithm
Modular arithmetic
Factoring
The Chinese Remainder Theorem
General Directions: You mus

Discrete Mathematics (550.171)
Homework 7 (Due Friday, April 05, 2013)
Objectives: The student will gain experience with
functions as relations
composition of functions
General Directions: You must show all work and document any assumptions to receive
f

First- and Second-Order Recurrence Relations
Recurrence Relation: A formula that species how each term in a sequence is produced from earlier
terms.
Examples:
(a) One of the most famous sequences of numbers is: 0, 1, 1, 2, 3, 5, 8, 13, 21, .
These numbers

Quantiers
Existential Quantier
Symbol:
Meaning: There exists, There is
Example:
There is an integer less than 5 that is even.
This can also be written:
There exists an x, a member of Z, such that x < 5 and x is even.
More formally:
x Z, x < 5 and x is ev

Rabins Method
To keep things simple, our message will only use the letters A through Z and a SP ACE for
separation. So A = 0, B = 1, . . . , Z = 26, SP ACE = 27. (Leading zeros are truncated.)
Alice wishes to send Bob the message NO which is put into numb

Number Theory Review
Self Test Problems
1. Find integers q and r such that 23 = 5q + r with 0 r < 5, and calculate 23 div 5 and 23 mod 5.
Solution: Notice that q is the quotient and r is the remainder. We want r to fall between 0 and 5,
so we need only to

Combinatorial Proof Sketch
May 12, 2011
Here is a combinatorial proof for the question on the quiz which was overly
dicult:
Consider the following:
n
i=0
x+i
i
=
n+x+1
n
Now consider the following expansion:
x+n
n
+
x+n1
n1
+ +
x+1
1
+
x
0
=
n+x+1
n
We wi

Direct Proofs
Direct proof of an if-then statement: (General steps)
(1) What do we know? State the hypothesis. If possible, rewrite with notation using letters
to represent variables.
(2) What are we trying to show? Write down the conclusion of the propos