Math 471
MIDTERM II INFORMATION
Fall 2014
WHEN: Monday, November 17, 2014, in class (no notes, books, calculators)
REVIEW SESSIONS:
Saturday, November 15 from 2:00 – 4:00 PM (in MH-615)
Sunday, November 16 from 12:00 – 2:00 PM (in MH-615)
EXTRA OFFICE HOURS:
Friday, November 14 from 10:00 AM – 12:00 PM and 4:00 – 7:00 PM
Saturday, November 15 from 4:00 – 6:00 PM
Sunday, November 16 from 8:00 PM – 12:00 midnight
Monday, November 14 from 10:00 – 12:00 NOON and 1:30 – 3:30 PM
COVERAGE: The midterm will cover the material discussed in lecture from Sections
7.1, 7.2, 7.4, 7.5, 11.1—11.5, 12.2.
STUDYING: Here is an overview of the topics we have covered. You should be
comfortable with all of the following words below:
Chapter 7: Recurrence Relations: homogeneous vs. non-homogeneous, constant
coefficient, Fibonacci sequence, Initial values, Towers of Hanoi, trial solution,
characteristic equation, general solution: repeated vs. non-repeated roots, particular
solution: polynomial and exponential cases, generating functions, using generating
functions to solve counting problems
Chapter 11: Graphs: simple, multigraph, general; Vertices, Edges, Adjacent, Adjacency
matrix, Incident, Loops, Connected, Connected components of a graph, Complete Graph,
Planar Graph, Degree, Degree sequence, Parity of degrees, Isomorphism, walks, open
and closed trails, paths, cycles, Eulerian trail, Ore property, Hamilton paths and cycles,
Bipartite graphs, Complete bipartite graphs, Trees, Spanning trees
Chapter 12: Planar graph, Euler’s formula, degree of a region, bounds on the edges in a
planar graph (including special cases on cycle lengths present), crossing number (not in
book)
IMPORTANT THEOREMS:
Theorem 7.1.1 (Formula for nth Fibonacci number)
Theorem 7.4.1 (Solution to homogeneous recurrence relation: distinct roots)
Theorem 7.4.2 (Solution to homogeneous recurrence relation: repeated roots)
Form of the trial solution to non-homogeneous recurrence relation: top of p. 249
Theorem 11.1.1 (Degree sequence must add to an even number)
Theorem 11.1.2 (Degree sequence is a structural property)
Theorem 11.2.2 (Criterion for a closed Eulerian trail)
Theorem 11.2.3 (Criterion for an open Eulerian trail)
Theorem 11.3.1 (Bridges are excluded from Hamilton cycles)
Theorem 11.3.2 (Ore property implies existence of Hamilton cycle)
Theorem 11.4.1 (Bipartite graphs contain only even-length cycles)
Theorem 11.5.2 (Number of edges in a tree)
Lemma 11.5.3 (Criterion for a bridge)
Theorem 11.5.4 (Trees contain no cycles)
Theorem 11.5.5 (Unique paths in trees)
Equation 12.5 and Groupwork #11 (Edge bounds on planar graphs)
THINGS TO BE ABLE TO DO:
Set up recurrence relations for physical problems
Solve both homogeneous and non-homogeneous recurrence relations via the method
of “trial solution” and/or “back-substitution” and/or “induction”
Prove relations involving recursively defined sequences by using induction (e.g.
Fibonacci relations were explored in homework/groupwork)
Determine the generating function for a sequence of numbers
Use generating functions to solve counting problems
Draw graphs with certain properties (e.g. Eulerian trails, Hamilton paths/cycles,
Bipartite, degree sequences, etc)
Prove that two graphs are or are not isomorphic to each other
Find or disprove the existence of an Eulerian trail and/or Hamilton path/cycle in a
graph
Decide (with proof) if a given graph is bipartite
Decide (with proof) if a given graph is a tree
Draw families of non-isomorphic graphs possessing a given property (e.g. all trees of
order 6, all graphs of order 4 containing a Hamilton cycle, etc.)
Perhaps do some basic proofs involving graph theory, similar to the level/difficulty
of those found in Section 11.5: (e.g. Theorems 11.5.2, 11.5.4, 11.5.5, etc.)
Decide if a given graph is planar or not, with proof
More generally, find (with proof) the crossing number of a given graph
ADVICE: I suggest reviewing the group work. If you have time, it might be best to try
re-working those exercises from scratch and looking up the answers afterwards. You will
not have time to re-do all of the homework, but you might try some of them again,
especially the ones that will assist you with the ``THINGS TO BE ABLE TO DO" listed
above. TRY TO RE-DO PROBLEMS FROM SCRATCH, RATHER THAN JUST
REVIEWING YOUR ALREADY-COMPLETED SOLUTIONS.
If you prefer, you can just try some extra problems from scratch. I've included a number
of random practice problems below that you can try to work. I will post solutions to them
as the midterm approaches.
It may also help you to discuss solutions to problems with other people. Explaining
yourself to someone else can be a tremendous benefit to both of you, especially with this
particular material. Often, there is more than one way to look at things, so it is useful to
hear each other's perspective on the problems at hand.
Finally, you can ask me questions as much as you want, and I will be happy to review or
pop-quiz a topic with you if you feel shaky. Basically, I'm here to help and I want
everybody to do well, so please don't be shy :=) !!
SAMPLE PROBLEMS:
1. Let fn denote the nth Fibonacci number (recall: f0 = 0 and f1 = 1). Show that for every
positive integer n, we have
fn+1 fn-1 – fn2 = (-1)n.
2. For each n ≥ 1, let f(n) be the number of subsets of {1,2,3,…,n} which contain no two
consecutive integers.
(a) Find f(1), f(2), and f(3).
(b) Find a recurrence relation for f(n) and use it to find f(8).
6. Find a recurrence relation for the number of ways to cover a 1 X n board with
monominoes (1 X 1), dominos (1 X 2), and triominos (1 X 3).
7. What are the initial conditions for the recurrence relation you found in Problem 6?
Use the relation to find the number of ways to cover a 1 X 9 board in Problem 6.
8. Solve each recurrence relation below:
(a): an = an-1 + 2an-2, with a0 = 2 and a1 = 7.
(b): an = -3an-1 – 3an-2 – an-3, with a0 = 1, a1 = -2, and a2 = -1.
(c): an = 5an-1 – 6an-2 + 7n, with no initial values given.
9. For n ≥ 1, let Qn be the graph whose vertices are labeled with bit strings of length n.
Two vertices are connected if and only if their bit string labels differ in exactly one place.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Draw Q1, Q2, Q3, and Q4.
How many vertices are there in Qn?
How many edges are there in Qn?
For which values of n is Qn bipartite?
For which values of n does Qn have a closed Eulerian trail?
For which values of n does Qn have a Hamilton cycle?
Determine the values of n such that Qn is planar.
(IGNORE FOR NOW) Determine the chromatic number of Qn.
10. Decide whether each graph below has an open Eulerian trail, closed Eulerian trail,
Hamilton cycle, and/or Hamilton path. Exhibit any of them that do exist, and prove why
the others cannot exist:
(a):
B
A
C
I
J
K
L
M
N
O
P
Q
D
F
H
G
(b):
R
V
S
W
T
X
E
U
(c):
A
F
E
J
G
B
H
I
D
C
11. Decide whether the pairs of graphs shown below are isomorphic or not. For those
that are, exhibit a valid isomorphism. For those that are not, find a structural difference
between the graphs:
(a):
A
B
E
F
H
G
1
2
5
6
8
D
C
7
4
3
(b):
1
A
B
2
5
3
D
E
C
4
(c):
1
A
B
F
E
4
C
D
3
6
5
2
12. Draw all isomorphism types of trees of order 6.
13. Find, with proof, the crossing numbers of the graphs in Figures 1-4 below:
FIGURE 1:
A
B
D
E
G
F
C
FIGURE 2:
I
H
J
M
N
K
L
FIGURE 3:
O
P
Q
V
U
R
T
S
FIGURE 4:
Yj
W
S
V
Z
T
U
X
14. (IGNORE FOR NOW) Find the chromatic number of each graph in Figures 1-4
above.
15. Determine whether each graph in Figures 1-4 above has (a) a closed Eulerian trail,
(b) an open Eulerian trail, (c) a Hamilton path, and (d) a Hamilton cycle.
16. Suppose that a valid codeword is an n-digit number in decimal notation containing an
even number of 0’s. Let an denote the number of valid codewords of length n.
(a) Show that the sequence {an} satisfies the recurrence relation an = 8an-1 + 10n-1, with
initial condition a1 = 9.
(b) Use generating functions to find an explicit formula for an.