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Discrete Mathematics, Spring 2004
Homework 7 Sample Solutions
6.1 #41.
How many edges are in an
n
cube?
Solution
. For concreteness, let’s use the bit string representation of the
n
cube, in which
vertices are connected by an edge if their corresponding strings di±er in exactly one position.
Let
b
n
denote the number of edges in the
n
cube. Then the ²rst few cases can be done
manually:
b
1
= 1,
b
2
= 4,
b
3
= 12. In general, one should notice that the
n
cube is obtained
from the (
n

1)cube in the following manner: take two copies of the (
n

1)cube (one copy
consisting of strings which start with a 0, the other consisting of strings which start with a
1), and connect with an edge those vertices whose strings match in bits 2 through
n
but di±er
in the ²rst bit. Thus we are adding 2
n
−
1
new edges, one for each vertex of the (
n

1)cube.
From this it follows that the recurrence relation for the sequence
b
n
follows the pattern
b
n
= 2
b
n
−
1
+ 2
n
−
1
, and we can solve by iteration:
b
n
= 2
b
n
−
1
+ 2
n
−
1
= 2(2
b
n
−
2
+ 2
n
−
2
) + 2
n
−
1
= 2
2
b
n
−
2
+ 2
·
2
n
−
1
= 2
2
(2
b
n
−
3
+ 2
n
−
3
) + 2
·
2
n
−
1
= 2
3
b
n
−
3
+ 3
·
2
n
−
1
=
···
= 2
n
−
1
b
1
+ (
n

1)
·
2
n
−
1
=
n
·
2
n
−
1
.
Note: one can use an alternate approach based on the methods of section 2. Notice that each
vertex has degree
n
, and there are 2
n
vertices in all. From Theorem 6.2.21 we know that the
sum of the degrees of the vertices is 2
b
n
, we conclude that
b
n
=
1
2
(number of vertices)(degree of each vertex) =
1
2
·
2
n
·
n
= 2
n
−
1
·
n.
1
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View Full Document 6.2 #15.
Draw a graph with four vertices having degree 1, 2, 3, and 4, or explain why no
such graph exists.
Solution
. There are a number of possible graphs satisfying these criteria. The following
is one possibility:
6.2 #18.
Draw a simple graph with Fve vertices having degrees 2, 2, 4, 4, and 4, or explain
why no such graph exists.
Solution
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This note was uploaded on 06/12/2011 for the course MATH 103 taught by Professor Wouters during the Spring '08 term at Wisc Oshkosh.
 Spring '08
 WOUTERS
 Math, Algebra

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