MATH20902 2 hours
THE UNIVERSITY OF MANCHESTER MANCHESTER
DISCRETE MATHEMATICS
26 May 2009 09:45 – 11:45
Answer
3 questions in Section Section A (30 marks in all) and TWO of the 3 questions in Section Section B (25 marks each). ALL
Electronic calculators may be used, provided that they cannot store text.
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P.T.O.
MATH20902 SECTIO SECTION N A
Answer
A1.
ALL
3 questions
Explain what it means to say that
• a graph G is connected • a graph G contains a cycle Prove that a connected graph on n vertices must have at least (n (n
A2.
− 1) edges.
[10 marks]
Given a graph G(V, E ), ), say what is meant by
• the degree sequence of G • the statement that G has an Eulerian cycle Suppose that G(V, E ) is a connected graph with n = V = 6 vertices and m = E = 13 edges: edges:
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(a) List List all the degree degree sequen sequences ces that G could have. (b) Can G have an Eulerian cycle? Justify your answer. [10 marks] Consider the following table, which lists the tasks required to complete a certain project, the time (in days) required to complete them and those tasks which are the immediate prerequisites.
A3.
Task ask
A B C D E F G
Time Time
Prer Prereequisi quisite tes s
5 2 3 6 1 8 4
None A B A B&D B&D C, E & F
(a) Draw a suitable suitable directed directed graph representing representing the project. (b) By finding a critical path through through the graph from part (a), find the shortest amount amount of time in which the project can be completed. [10 marks]
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MATH20902 SECTION B
Answer
2
of the 3 questions
B4.
(a) Explai Explain n what what is meant meant by the notation notation x = [a0 : a1 , a2, a3 ] and by the term simple continued fraction fraction (SCF). (b) Characterize Characterize those real numbers numbers that have repeating infinite simple continued continued fraction representations of the form x = [3 : a, b, a, b, a, b,... ] and hence, or otherwise, find the SCF for x =
√ 12.
√ 12 such that q < 100 and
(c) Find a rational approxi approximation mation p/q for
√
12
−
p 1 < . q 1000
Now consider an infinite sequence of positive integers a0, a1 , . . . , a j , . . . and use it to define two further sequences sequences p j and q j according to the recursion relations p0 = a0 p1 = a0 a1 + 1 p j = a j p j −1 + p j −2
for j
≥2
q 0 = 1 q 1 = a1 q j = a j q j −1 + q j −2
for j
≥2
(B4.1)
(d) Prove Prove that that if p if p j and q j are as defined in (B4.1), then p j q j holds for all j
−
p j −1 ( 1) j +1 = q j −1 q j q j −1
−
≥ 1. [25 marks]
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MATH20902 B5.
Consider the following graph G(V, E ) A
B
C
F
E
D
(a) Define a Hamiltonian cycle and find one in the graph G above. (b) Explain Explain how to construct construct the closure of a graph and sketch the closure [G [G] of the graph above, indicating which edges, if any, you have added. (c) State Kuratowski Kuratowski’s ’s Theorem. (d) Is the graph above above planar? planar? Explai Explain n wh why y or wh why y not. (e) What What is the smallest smallest number number of edges that one can remove remove to conv convert the graph graph above above into into a planar one? Justify your answer by providing a planar diagram. [25 marks]
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MATH20902 In a particu particularl larly y friendl friendly y and upup-to-d to-date ate sectio section n of Manches Manchester’ ter’ss Norther Northern n Quarter, Quarter, the neighbours are trying to choose channels for their WiFi routers in such a way that they don’t interf interfere ere with each other. other. If we label their flats with letters letters A–G then the distances (in meters) between them are given by the table below.
B6.
A B C D E F G
B
85 13 7 1 6 5 12 3 3 9 16 4 1 3 2 10 5 7 5 13 4 1 9 1
C
D
E
F
G
2 05 1 1 7 1 71 235 92 201 252 223 298 177
(a) Say Say what is meant meant by the terms
• a k-colouring of a graph G; • the chromatic number χ(G) of a graph G. (b) Suppose that good wireless reception reception requires that neighbours neighbours within 150 meters of each other should should use differe different nt channels channels.. What is the smallest smallest nu numbe mberr of channe channels ls that the neigh neighbours bours can use? Justify your answer rigorously. (c) For a graph G(V, E ), ), say what is meant by
• a set of palettes for G; • a list colouring with respect to a set of palettes; ch(G). • the choosability ch(G (d) Prove Prove that that if G if G(V, E ) is a graph on n vertices vertices then χ(G)
ch(G) ≤ n. ≤ ch(G [25 marks]
END OF EXAMINATION EXAMINATION PAPER
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