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 10 minutes reading time
This booklet contains 10 pages.
The University of Sydney
School of Mathematics and Statistics
Solutions to Tutorial 6 (Week 7)
MATH2069/2969: Discrete Mathematics and Graph Theory
Semester 1, 2015
1. Use generating functions to solve the following recurrence relations. (Hint: you
can re
The University of Sydney
School of Mathematics and Statistics
Tutorial 6 (Week 7)
MATH2069/2969: Discrete Mathematics and Graph Theory
Semester 1, 2015
More dicult questions are marked with either * or *. Those marked * are at the level
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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-homogeneous recurrence relations:
(a) an = 3an1 + 4an2 12n 2 fo
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 * or *. Those marked * are at the level
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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 n 0,
(a) n3 + 5n is a multiple of 3 (i.e. n3 + 5n = 3
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 algorithm to nd a spanning tree of the Petersen
graph.
The University of Sydney
School of Mathematics and Statistics
Tutorial 4 (Week 5)
MATH2069/2969: Discrete Mathematics and Graph Theory
Semester 1, 2015
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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 15 dierent avours, which it sells in bags of 10. If
the
The University of Sydney
School of Mathematics and Statistics
Tutorial 3 (Week 4)
MATH2069/2969: Discrete Mathematics and Graph Theory
Semester 1, 2015
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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 ornaments to put on your mantelpiece.
(a) If you want to use
The University of Sydney
School of Mathematics and Statistics
Tutorial 2 (Week 3)
MATH2069/2969: Discrete Mathematics and Graph Theory
Semester 1, 2015
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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, dene a function fA : X cfw_0, 1 by setting
fA (x) = 1 i
The University of Sydney
School of Mathematics and Statistics
Tutorial 1 (Week 8)
MATH2069/2969: Discrete Mathematics and Graph Theory
Semester 1, 2015
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The University of Sydney
School of Mathematics and Statistics
Solutions to Tutorial 1 (Week 8)
MATH2069/2969: Discrete Mathematics and Graph Theory
Semester 1, 2015
1. In each part, determine whether the two pictures represent the same isomorphism
class o
The University of Sydney
School of Mathematics and Statistics
Tutorial 2 (Week 9)
MATH2069/2969: Discrete Mathematics and Graph Theory
Semester 1, 2015
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8038/8039
Semester 1, 2014
Revision Tutorial (Week 13) (2014 Exam Paper)
Faculties of Arts, Economics, Education,
Engineering and Science
MATH2069/2969: Discrete Mathematics and Graph Theory
Lecturer: Alexander Molev
Time allowed: 2 hours, plus 10 minutes
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:
F(z)+C(z)=1 I 22 I 322 I 723 I 1724 I
We obtain each c
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},
A3 : {n E N| 100 g n S 999, n is palindromic},
and t
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 are not allowed to be assigned the same time. From a m
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 where it started. (We saw such a walk in Example 2.6.)
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, - - - ,nn} such that v,- is adjacent to 024.1 for 2' : 1, - -
The University of Sydney
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Tutorial 4 (Week 11)
MATH2069/2969: Discrete Mathematics and Graph Theory
Semester 1, 2015
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The University of Sydney
School of Mathematics and Statistics
Solutions to Tutorial 3 (Week 10)
MATH2069/2969: Discrete Mathematics and Graph Theory
Semester 1, 2015
1. For vertices v and w in a connected weighted graph, d(v, w) denotes the minimum
weight
The University of Sydney
School of Mathematics and Statistics
Tutorial 3 (Week 10)
MATH2069/2969: Discrete Mathematics and Graph Theory
Semester 1, 2015
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The University of Sydney
School of Mathematics and Statistics
Solutions to Tutorial 2 (Week 9)
MATH2069/2969: Discrete Mathematics and Graph Theory
Semester 1, 2015
1. Determine whether each of the following sequences is the degree sequence of a
graph. If
The University of Sydney
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Tutorial 1 (Week 2)
MATH2069/2969: Discrete Mathematics and Graph Theory
Semester 1, 2015
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The University of Sydney
School of Mathematics and Statistics
Tutorial 3 (Week 4)
MATH2069/2969: Discrete Mathematics and Graph Theory
Semester 1, 2013
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