MATH 2602 REVIEW 3 INSTRUCTIONS for Final: The final exam is

MATH 2602 REVIEW 3
INSTRUCTIONS for Final:
The final exam is cumulative. USE also Reviews 1 & 2. In addition to the
problems in the Reviews 1, 2 & 3 look at Tests 1 & 2, and quizzes.
In principle, topics not covered in the review problems or review sessions, but
covered in lectures, may appear on the final exam.
No books, calculators/computers are permitted. A two-sided standard-sized formula sheet is allowed. The formula sheet can contain theorems, definitions, formulas, etc..., but cannot have solutions to specific problems. Write your name on the
formula sheet and turn it in with the exam.
You must show your work to get full credit. This means you must
explain, give reasons for... why you arrive at the answers.
The Georgia Tech Honor Code is in effect for this exam.
1. Complexity
(1) Find a function f which is the product of functions in the list
log n, na , an , n!, nn
(where a is a constant) such
that f has the same order as
√
5
4
2
22
(a) g(n) = n − 5n + n + n3 + 1
(b) g(n) = n log3 (n4 + 1) + 3n2
(c) g(n) = 2n + (n − 1)n + n!
3n5 + 2n3 log5 n + 7
p
(d) g(n) =
4
2n + 3 n(log n)20 + n13 + 9
n
(e) g(n) = 4n + n4
(f) g(n) = (n/2)n + nn/2
(2) Show that log(n!) = O(n log n).
(3) Problems 2 and 17 of 8.2
2. Graph
(1) Problems 6 and 13 of page 301.
(2) Let e be an edge of K5 and G = K5 − {e}.
(a) What is the chromatic number of G?
(b) Is G planar?
(c) Does G contain a Hamiltonian cycle?
(d) Does G contain an Eulerian circuit?
(e) Write down the Laplacian matrix, and calculate the number of spanning trees (in the form of the determinant of a matrix) of G.
(f) Calculate the actual number of spanning trees of Kn − {e}, where e
is an edge of Kn .
solution
(f ) Kn has m = n2 = n(n − 1)/2 edges.
1
2
There are A = nn−2 spanning trees of Kn . Each spanning tree
has B = n − 1 edges. The total number of edges (with repetition)
in all A spanning trees is AB = (n − 1)nn−2 . Since there are m
edges, each edge must appear in exactly
C=
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
AB
(n − 1)nn−2
=
= 2nn−3 .
n
m
2
Thus, the number of spanning trees of Kn not containing e is
A − C = nn−2 − 2nn−3 . This is the number of spanning trees of
Kn − e.
Suppose each of the two graphs F1 and F2 has 7 vertices and 20 edges.
Show that F1 and F2 are isomorphic.
(i) For what n does the graph K4,n
a) have an Eulerian cycle?
b) have a Hamiltonian cycle?
d) have a Hamiltonian path?
c) a planar embedding?
ii) How many Hamiltonian cycles does K6,6 have? (You can assume that
all Hamiltionan cycles begin at a fixed vertex.)
Problem 9 of page 310.
Show that the hypercube Q4 is not planar. (Hint: Q4 is bipartite).
Solution. In lecture, we proved that if a connected graph is
bipartite and planar, then it has a vertex of degree < 4. The
graph Q4 is connected, bipartite, and has no vertex of degree < 4
(every vertex of Q4 has degree 4). Hence Q4 is not planar.
Problem 5 of section 13.1.
A connected plane graph has 7 regions. The numbers of edges on the
boundaries of the regions are respectively 3,3,3,3,3,4,5. Find the number of
vertices of the graph.
Problems 7 of page 336.
Suppose T is a tree with degree sequence 5, 5, 5, 4, 4, 3, 3 plus several 2’s
and 1’s. Find the number of leaves of T .
Problems 11 of page 409.
There are 40 chemicals {1, 2, . . . , 40}. If |i − j| < 5, then i and j cannot
safely be stored in the same room. Determine the minimum number of
rooms needed for safe storage of the chemicals.
(Translate this into a graph problem. What graph G? What is the
maximum degree of G? What is the size of the maximum complete subgraph
in G? What is the chromatic number of G?)
Let G be a weighted complete graph with vertex set V (G) = {1, 2, .., 6}
and the weight of the edge between vertices i and j equal to i2 − ij + j 2 .
Find the minimum weight spanning tree in G and the shortest paths from
vertex 3 to all the other vertices of G.