Fall 2009
CS131
– Combinatorial Structures
Homework 10
Homework 10, due Dec 8
You must prove your answer to every question.
Do not rely only on the homework for exercise: there are several selfcheck ex
ercises of the easier kind in the book, try to solve them, too!
Problem 1.
(a) (5pts) Which expression grows faster:
(
n
4
)
or
(
n
+
3
n
)
?
Solution.
We have
n
4
=
n
(
n

1)(
n

2)(
n

3)
4!
,
n
+
3
n
=
n
+
3
3
=
(
n
+
1)(
n
+
2)(
n
+
3)
3!
.
The first expression is a polynomial of degree 4, the second one a polynomial of
degree 3, so the first expression grows faster.
(b) (10pts) Show that the expression
(
3
n
n
)
grows exponentially.
Solution.
We have
3
n
n
=
3
n
(3
n

1)
···
(2
n
+
1)
n
·
(
n

1)
···
1
.
Every factor of the top is at least twice larger than any factor at the bottom. So
the whole expression is
2
n
, since we can write it as
3
n
n
3
n

1
n

1
···
2
n
+
1
1
,
each of the
n
factors of which is greater than 2. For an upper bound, we know
(
3
n
n
)
2
3
n
=
8
n
.
Problem 2.
(15 pts) Decide which of the following graphs has a Hamilton cycle.
The answer only counts if you prove your statements.
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Fall 2009
CS131
– Combinatorial Structures
Homework 10
a
1
a
2
a
3
a
4
a
5
b
1
b
2
b
3
b
4
b
5
c
1
c
2
c
3
c
4
c
5
d
1
d
2
d
3
d
4
d
5
A
1
A
2
A
3
A
4
A
5
B
1
B
2
B
3
B
4
B
5
Solution.
For the first graph, a Hamilton cycle is given by
a
1
,
b
1
,
d
3
,
c
3
,
c
2
,
c
1
,
c
5
,
c
4
,
d
4
,
b
2
,
d
5
,
b
3
,
d
1
,
b
4
,
d
2
,
b
5
,
a
5
,
a
4
,
a
3
,
a
2
.
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 Fall '11
 FredPhelps
 Math, hamiltonian cycle, Hamiltonian path, Bipartite graph, combinatorial structures

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