8038/8039
Semester 1, 2014
Faculties of Arts, Economics, Education,
Engineering and Science
MATH2069/2969: Discrete Mathematics and Graph Theory
Lecturer: Alexander Molev
Time allowed: 2 hours, plus 1
The University of Sydney
School of Mathematics and Statistics
Solutions to Tutorial 4 (Week 11)
MATH2069/2969: Discrete Mathematics and Graph Theory
Semester 1, 2015
1. Apply the Breadth-First Search
The University of Sydney
School of Mathematics and Statistics
Tutorial 4 (Week 5)
MATH2069/2969: Discrete Mathematics and Graph Theory
Semester 1, 2015
More dicult questions are marked with either * o
The University of Sydney
School of Mathematics and Statistics
Solutions to Tutorial 3 (Week 4)
MATH2069/2969: Discrete Mathematics and Graph Theory
Semester 1, 2015
1. A factory makes jelly beans of 1
The University of Sydney
School of Mathematics and Statistics
Tutorial 3 (Week 4)
MATH2069/2969: Discrete Mathematics and Graph Theory
Semester 1, 2015
More dicult questions are marked with either * o
The University of Sydney
School of Mathematics and Statistics
Solutions to Tutorial 2 (Week 3)
MATH2069/2969: Discrete Mathematics and Graph Theory
Semester 1, 2015
1. Suppose you have 7 dierent ornam
The University of Sydney
School of Mathematics and Statistics
Tutorial 2 (Week 3)
MATH2069/2969: Discrete Mathematics and Graph Theory
Semester 1, 2015
More dicult questions are marked with either * o
The University of Sydney
School of Mathematics and Statistics
Solutions to Tutorial 1 (Week 2)
MATH2069/2969: Discrete Mathematics and Graph Theory
Semester 1, 2015
1. Given a set X and a subset A X,
The University of Sydney
School of Mathematics and Statistics
Tutorial 1 (Week 2)
MATH2069/2969: Discrete Mathematics and Graph Theory
Semester 1, 2015
More dicult questions are marked with either * o
The University of Sydney
School of Mathematics and Statistics
Tutorial 3 (Week 4)
MATH2069/2969: Discrete Mathematics and Graph Theory
Semester 1, 2013
More dicult questions are marked with either * o
The University of Sydney
School of Mathematics and Statistics
Tutorial 2 (Week 3)
MATH2069/2969: Discrete Mathematics and Graph Theory
Semester 1, 2013
More dicult questions are marked with either * o
The University of Sydney
School of Mathematics and Statistics
Solutions to Tutorial 5 (Week 6)
MATH2069/2969: Discrete Mathematics and Graph Theory
Semester 1, 2013
1. Solve the following non-homogene
The University of Sydney
School of Mathematics and Statistics
Solutions to Tutorial 2 (Week 3)
MATH2069/2969: Discrete Mathematics and Graph Theory
Semester 1, 2013
1. Suppose you have 7 dierent ornam
The University of Sydney
School of Mathematics and Statistics
Tutorial 4 (Week 5)
MATH2069/2969: Discrete Mathematics and Graph Theory
Semester 1, 2013
More dicult questions are marked with either * o
The University of Sydney
School of Mathematics and Statistics
Solutions to Tutorial 1 (Week 2)
MATH2069/2969: Discrete Mathematics and Graph Theory
Semester 1, 2013
1. Given a set X and a subset A X,
The University of Sydney
School of Mathematics and Statistics
Solutions to Tutorial 3 (Week 4)
MATH2069/2969: Discrete Mathematics and Graph Theory
Semester 1, 2013
1. A factory makes jelly beans of 1
The University of Sydney
School of Mathematics and Statistics
Tutorial 6 (Week 7)
MATH2069/2969: Discrete Mathematics and Graph Theory
Semester 1, 2013
More dicult questions are marked with either * o
The University of Sydney
School of Mathematics and Statistics
Tutorial 5 (Week 6)
MATH2069/2969: Discrete Mathematics and Graph Theory
Semester 1, 2013
More dicult questions are marked with either * o
The University of Sydney
School of Mathematics and Statistics
Solutions to Tutorial 4 (Week 5)
MATH2069/2969: Discrete Mathematics and Graph Theory
Semester 1, 2015
1. Prove by induction that, for all
The University of Sydney
School of Mathematics and Statistics
Tutorial 5 (Week 6)
MATH2069/2969: Discrete Mathematics and Graph Theory
Semester 1, 2015
More dicult questions are marked with either * o
The University of Sydney
School of Mathematics and Statistics
Solutions to Tutorial 5 (Week 6)
MATH2069/2969: Discrete Mathematics and Graph Theory
Semester 1, 2015
1. Solve the following non-homogene
78 TOPICS IN DISCRETE MATHEMATICS
Example 3.5. According to these rules, the sum of
F(z):z+22+223+3z4+- a11dC(z):1 I z I 222 I 523 I 14241 I
is just obtained by adding the coefficients term-byterin:
10 TOPICS IN DISCRETE MATHEMATICS
you are using the fact that it is the disjoint union ofsubsets A1, A2, A3, where
A1 : {n E N| 1 S n S 9, n is palindromic},
A2 = {n E N] 10 g n S 99, n is palindromic
72 INTRODUCTION TO GRAPH THEORY
In applications to scheduling, the vertices represent the different tasks, the
colours represent the available times for them, and the edges record which
pairs of tasks
CHAPTER 2. SPECIAL WALKS IN GRAPHS 31
There is an easy variant of Theorem 2.6 for the case of an Eulerian trail,
which is a walk that uses every edge exactly once but nishes at a different
vertex from
CHAPTER 1. FIRST PROPERTIES OF GRAPHS 17
Denition 1.31. For any graph G, a path in G is a subgraph of G which
is isomorphic to B, for some n: that is, a collection of distinct vertices
{vb 02, - - -
The University of Sydney
School of Mathematics and Statistics
Tutorial 4 (Week 11)
MATH2069/2969: Discrete Mathematics and Graph Theory
Semester 1, 2015
More dicult questions are marked with either *