Download CC9066 Semester 1, 2016 Faculties of Arts, Economics, Education

CC9066
Semester 1, 2016
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 5 pages.
This paper comprises 6 questions of equal value. Each question is divided into several parts.
Questions 1, 2, 4, 5 are the same for MATH2069 and MATH2969. For
questions 3, 6 this paper contains both the mainstream-level MATH2069
question and the (completely different) advanced-level MATH2969
question. You may ONLY answer the questions for the unit you are
enrolled in.
If you can’t solve one part of a question, you can still assume the result
in doing later parts.
No notes or books are allowed. Approved calculators are permitted.
CC9066
Semester 1, 2016
page 2 of 5
1. Give explicit numerical values for your answers in parts (c) and (d).
A lottery ticket is a table of 30 numbers 1, 2, . . . , 30 with 5 chosen numbers crossed
out.
(a) What is the total number of different lottery tickets?
(b) A draw is a 5-subset of the set {1, 2, . . . , 30}. A lottery ticket wins if it
contains at least 3 numbers from the draw. Given a draw, what is the
number of winning tickets?
(c) A discrete mathematics student always crosses out only Stirling numbers.
List the Stirling numbers contained in the set {1, 2, . . . , 30} and hence calculate the number of such lottery tickets.
(d) Another student first crosses out two consecutive numbers, then chooses
three consecutive numbers to cross out. What is the number of such lottery
tickets?
2.
(a) Find the general solution of the recurrence relation
bn = −2 bn−1 + 8 bn−2 where n > 2.
(b) Use the previous part or otherwise to explain why the non-homogeneous
recurrence relation an = −2 an−1 + 8 an−2 + 3 (−4)n where n > 2, does not
have a particular solution in the form pn = C (−4)n .
(c) Find the solution of the recurrence relation in part (b) satisfying the initial
conditions a0 = 3 and a1 = −14.
(d) Suppose that a sequence an is a solution of the recurrence relation in part
(b). Write down a non-homogeneous linear recurrence relation satisfied by
the sequence cn = an+5 with n > 0.
turn to page 3
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page 3 of 5
3. This question is for MATH2069 students only.
(a) For any sequence an define another sequence cn by the formula
cn = na0 +(n−1)a1 +· · ·+2an−2 +an−1 . Find an expression for the generating
function C(z) of the sequence cn in terms of the generating function A(z) of
the sequence an .
(b) Use the previous part or otherwise to explain why the generating function
D(z) of the sequence dn = n 30 + (n − 1)31 + · · · + 2 · 3n−2 + 3n−1 is found by
z
.
D(z) =
(1 − 3z)(1 − z)2
(c) Hence find an explicit formula for the sequence dn defined in part (b).
3. This question is for MATH2969 students only.
(a) The generating function D(z) for a sequence dn is given by
D(z) =
1 + 3z
.
(1 − 4z)3
Give an explicit formula for dn .
(b) Consider the sequence bn defined by bn = (−3)n d0 + (−3)n−1 d1 + · · · + dn for
all n > 0. Find an explicit formula for bn .
−2an−1 + 2an−2 − 15an−3
(c) A sequence an satisfies the recurrence relation an =
n
for n > 3 with a0 = 3, a1 = −6 and a2 = 9. Find a closed formula for the
generating function A(z).
turn to page 4
CC9066
4.
Semester 1, 2016
page 4 of 5
(a) Give the definition of isomorphism between two graphs G = (V, E) and
G′ = (V ′ , E ′ ).
(b) The two pictures represent graphs G and G′ :
G
G′
Determine whether the pictures represent isomorphic graphs. Justify your
answer.
(c)
(i) State the Hand-shaking Lemma.
(ii) A regular graph of degree 5 has 12 vertices. What is the number of
edges?
(d) Determine whether or not the following graph is Hamiltonian. Justify your
answer.
b
c
a
d
e
g
5.
f
j
i
h
(a) Find a walk which solves the Travelling Salesman Problem for the following
weighted graph. Justify your answer.
c
b
3
2
a
8
7
6
4
d
2
3
4
e
10
f
(b) Find a minimal spanning tree for this weighted graph.
(c) Consider the trees with 7 vertices {1, 2, 3, 4, 5, 6, 7} which have exactly 4
leaves.
(i) Write down all possible degree sequences of these trees.
(ii) Find 4 representatives of isomorphism classes of such trees and explain
why they are pairwise non-isomorphic.
(iii) Prove that any such tree is isomorphic to one of the four trees in the
previous part.
turn to page 5
CC9066
Semester 1, 2016
page 5 of 5
6. This question is for MATH2069 students only.
(a) The graph H is given by the picture
(i) What is the maximum possible value of the chromatic number χ(H)
as provided by Brooks’ Theorem? Justify your answer.
(ii) What is the exact value of χ(H)? Justify your answer.
(iii) What are the possible values of the edge chromatic number χ′ (H) as
provided by Vizing’s Theorem?
(iv) What is the exact value of χ′ (H)? Justify your answer.
(b)
(i) Find all graphs G whose chromatic polynomials have the form
PG (t) = t4 − 4t3 + bt2 + ct for some integers b and c.
(ii) Hence find all values of b and c such that t4 −4t3 +bt2 +ct is a chromatic
polynomial.
6. This question is for MATH2969 students only.
(a) For any m > 3 the graph Hm has 2m vertices (a1 , a2 , . . . , am ) which are mtuples of coordinates ai equal to 0 or 1. Two vertices are adjacent if the
m-tuples differ at exactly 3 coordinates.
(i) What is the chromatic number of Hm ? Give your reasons.
(ii) What is the edge chromatic number of Hm ? Give your reasons.
(b) A graph has 5 vertices and its chromatic number is 4.
(i) List all possible chromatic polynomials of this graph.
(ii) For each polynomial P (t) in your list give an example of a graph G
such that PG (t) = P (t).
This is the end of the examination paper