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CPSMA 3913 - Discrete Mathematics
Name
Assignment #5
due: May 2, 202
COMPLETE THIS SECTION FIRST:
I.
Before completing the questions, you will first calculate a few values. These values will be used
throughout the question to “randomize” questions. Clearly write your values for a, b and c at the top
of your assignment.
II.
You may need to find a variation of one of your values. For example, since my value of c is 6 then, I
would replace [c+2] with 8.
1. The Value of a=3, b=5, and c=6 ….. (i)
2. The Value of a=4, b=12, and c=3 ….(ii)
3. The Value of a=4, b=11, and c=8 ….(iii)
QUESTIONS:
1. (5 points) This algorithm shuffles the values in the sequence a1,...,an.
Input: a, n Output: a
(shuffled)
shuffle(a, n){
for i = 1 to n − 1
swap(ai, arand(i,n)) }
Trace the above algorithm for input: [a], [b], [c], 101. Include a table showing how each variable changes
at each step. Assume that the values of rand are:
rand(1,4)=2 rand(2,4)=4
rand(3,4)=3
2. (5 points) Write an algorithm that returns the location of the smallest value in the sequence s1,...,sn. Use
pseudocode and follow the guidelines discussed in class.
3. (5 points) Show that f(x) = [a]x + 9 is O(x). Clearly state your C and k and show that |f(x)| ≤ C|x| for x > k.
4. (5 points) Select a theta notation from the following: Θ(1), Θ(lgn), Θ(n), Θ(nlgn), Θ(n2), Θ(n3), Θ(2n), or
Θ(n!); for the number of times x + 1 is executed in the following segment. Show work! for i = 1 to 2n for j
= 1 to i x = x + 1
5. (5 points) There is a group of 5 people, Al, Bob, Cam, Dan, and Euclid. Some in the group might be friends.
Some in the group might not be friends. Can each person be friends with exactly [a] people in the group?
If so, then represent this situation with a graph. If not, explain why not.
6. (10 points) Consider the following two graphs. G1 is given as an ordered pair of sets and G2 is given with a
picture.
G1 = (V,E),
V = {a,b,c,d,e},
E = {{a,b},{a,c},{a,e},{b,d},{b,e},{c,d}}
G2 =
a. Does the function f : V1 → V2 given by the following table define an isomorphism between G1 and G2?
Why or why not?
x
a b c d e
f(x) v4 v5 v1 v3 v2
b. Give an isomorphism g between G1 and G2.
7. (3 points) For each graph, determine if it is bipartite or not. For those graphs that are bipartite, label each
vertex as being in either set A or in set B.
8. (5 points) Draw a planar representation of K4, the complete graph on 4 vertices.
9. (5 points) Draw the complete bipartite graph K[a+1],3.
10. (5 points) What is the chromatic number of the complete bipartite graph K[a+1],3.
11. (5 points) Is it possible for a connected planar graph to have 6 vertices, 10 edges, and [b+1] faces? Explain.
12. (5 points) A convex polyhedron contains 12 faces. Seven of the faces are triangles and four of the faces are
quadrilaterals and the shape of one of the faces is unknown. The polyhedron has 11 vertices total. How
many sides does the last face have?
13. (5 points) Draw a graph with chromatic number [a+2].
14. (5 points) A graph with chromatic number [b+5] cannot be planar. Explain why.
15. (5 points) For which n does the graph Kn contain a closed eulerian trail? Explain.
16. (5 points) Give an example of a graph that has a Hamiltonian cycle but does not have a closed eulerian
trail. Be creative! Your graph should not match any other student’s graph.
17. (5 points) Give an example of a graph that does not have a Hamiltonian cycle but does have a closed
eulerian trail. Be creative! Your graph should not match any other student’s graph.
18. (3 points) Which of the following graphs are trees?
19. (5 points) Draw the following tree as a rooted tree with vertex [a] as the root.
20. (4 points) Consider the rooted tree given below.
a. Which vertex is the root?
b. Which vertices are terminal?
c. Which vertices are children of b?
d. Which vertex is the parent of m?
e. Which vertices are siblings of r?
f. Which vertices are ancestors of j?
g. Which vertices are descendants of c?
h. Draw the subtree rooted at j.
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