NANYANG TECHNOLOGICAL UNIVERSITY SEMESTER II EXAMINATION MH 1301 DISCRETE MATHEMATICS TIME ALLOWED: 2 HOURS

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1 NANYANG TECHNOLOGICAL UNIVERSITY SEMESTER II EXAMINATION MH 1301 DISCRETE MATHEMATICS May 2016 TIME ALLOWED: 2 HOURS INSTRUCTIONS TO CANDIDATES 1. This examination paper contains FIVE (5) questions and comprises FIVE (5) printed pages. 2. Answer all questions. The marks for each question are indicated at the beginning of each question. 3. Answer each question beginning on a FRESH page of the answer book. 4. This IS NOT an OPEN BOOK exam. 5. Candidates may use calculators. However, they should write down systematically the steps in the workings.

2 Question 1 (20 marks) How many ways are there to arrange the letters of the word CORRECT, if (a) it starts with letter C or ends with letter R? (10 marks) There are 6! = 360 strings start with letter C, 6! = 360 2! 2! strings end with letter R, and 5! = 120 strings that both start and end with letter C. By the inclusion-exclusion principle, the final answer is = 600. (b) no two same letters are consecutive to each other? (10 marks) Only letter C and R appear twice in the word CORRECT. There are in total 6! = 360 strings that contain CC, 360 strings that 2! contain RR, and 5! = 120 strings that contain both CC and RR. By the inclusion-exclusion principle, there are = 600 strings that contain either CC or RR. 7! On the other hand, there are in total = 1260 strings in total, so there 2!2! are = 660 strings that do not contain any two consecutive same letters. The final answer is

3 Question 2 (45 marks) Answer each of the following questions (NO justification needed): (a) How many non-isomorphic simple undirected graphs with 3 vertices are there? (b) How many non-isomorphic trees with 4 vertices are there? (c) Can a simple graph with 6 vertices have degree sequence ? If yes, draw such a graph. (d) What is the length of the longest simple path in K 4,10? (e) What is the chromatic number of C 10? (f) What is the largest value of n for which K 6,n is planar? (g) If G is a connected planar graph with 12 regions and 20 edges, how many vertices does the graph have? (h) What is the largest distance between any two vertices in K 100,100? (i) Given a rooted tree with 1000 vertices, what are its minimum height and maximum height? (j) List all positive integers n such that K n has an Euler circuit. (k) Yes or no: Is every tree bipartite? (l) Yes or no: Does every planar graph have to be connected? (m) Yes or no: If two trees have the same number of vertices, the same number of edges, and the same degree sequence, are these two trees always isomorphic? (n) Yes or no: Does every a graph that contains an Euler circuit also contains a Hamilton circuit. (o) Yes or no: Does every a graph that contains a Hamilton circuit also contains an Euler circuit. 2

4 (a) 4 (b) 2 (c) Yes (d) 8 (e) 2 (f) 2 (g) 10 (h) 2 (i) minimum height 1, maximum height 999. (j) n odd (k) Yes (l) No (m) No (n) No (o) No 3

5 Question 3 (15 marks) An n-cube graph, denoted by Q n, is a graph that has vertices representing the 2 n bit strings of length n. Two vertices are connected by an edge if and only if the bit strings that they represent differ in exactly one bit position. (a) Draw Q 1, Q 2, Q (a) Q 1 (b) Q 2 (c) Q 3 (b) How many edges does Q n have? Q n has n2 n 1 edges. (c) For what values of n does Q n have an Euler circuit? And an Euler path? Q n has an Euler circuit if and only if n is even. Q n has an Euler path if and only if n is even or n = 1. 4

6 Question 4 (10 marks) Find a minimal spanning tree for the following weight graph using Prim s algorithm. a b c d e f g h i 3 7 Prim s algorithm adds the edges in the following order: {a, b} of weight 2, {b, e} of weight 1, {d, e} of weight 1, {e, f} of weight 2, {b, c} of weight 3, {e, h} of weight 4, {h, g} of weight 3, {f, i} of weight 6. The weight of the minimum spanning tree is 22. The resulting tree is as following. a b c d e f g h i 3 5

7 Question 5 (10 marks) (a) Find the preorder, inorder and postorder traversal of the following rooted tree. a b c d e f g h i j k The preorder traversal is a, b, e, f, j, k, c, d, g, h, i the inorder traversal is e, b, j, f, k, a, c, g, d, h, i the postorder traversal is e, j, k, f, b, c, g, h, i, d, a 6

8 (b) A rooted tree T has preorder traversal as a, b, c, d, e, inorder traversal as b, a, d, c, e, postorder traversal as b, d, e, c, a. Draw this tree or prove that such a tree does not exist. The tree is as following: a b c d e END OF PAPER 7

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