CS 70
Discrete Mathematics for CS
Fall 2006
Lecture 16
Hypercubes
Recall that the set of all
n
bit strings is denoted by
{
0
,
1
}
n
. The
n
dimensional hypercube is a graph whose
vertex set is
{
0
,
1
}
n
(i.e. there are exactly 2
n
vertices, each labeled with a distinct
n
bit string), and with
an edge between vertices
x
and
y
iff
x
and
y
differ in exactly one bit position. i.e. if
x
=
x
1
x
2
...
x
n
and
y
=
y
1
y
2
...
y
n
, then there is an edge between
x
and
y
iff there is an
i
such that
∀
j
n
=
i
,
x
j
=
y
j
and
x
i
n
=
y
i
.
There is another equivalent recursive definition of the hypercube:
The
n
dimensional hypercube consists of two copies of the
n

1dimensional hypercube (the 0subcube and
the 1subcube), and with edges between corresponding vertices in the two subcubes. i.e. there is an edge
between vertex
x
in the 0subcube (also denoted as vertex 0
x
) and vertex
x
in the 1subcube.
Claim:
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This note was uploaded on 11/03/2008 for the course CMPSC 360 taught by Professor Haullgren during the Fall '08 term at Penn State.
 Fall '08
 HAULLGREN

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