Massachusetts Institute of Technology
6.042J/18.062J, Spring ’10
: Mathematics for Computer Science
March 10
Prof. Albert R. Meyer
revised March 8, 2010, 683 minutes
Solutions
to
In-Class
Problems
Week
6,
Wed.
Problem
1.
For each of the following pairs of graphs, either deﬁne an isomorphism between them, or prove
that there is none. (We write
ab
as shorthand for
a
—
b
.)
(a)
G
1
with
V
1
=
{
1
,
2
,
3
,
4
,
5
,
6
}
, E
1
=
{
12
,
23
,
34
,
14
,
15
,
35
,
45
}
G
2
with
V
2
=
{
1
,
2
,
3
,
4
,
5
,
6
}
, E
2
=
{
12
,
23
,
34
,
45
,
51
,
24
,
25
}
Solution.
Not isomorphic:
G
2
has a node, 2, of degree 4, but the maximum degree in
G
1
is 3.
(b)
G
3
with
V
3
=
{
1
,
2
,
3
,
4
,
5
,
6
}
, E
3
=
{
12
,
23
,
34
,
14
,
45
,
56
,
26
}
G
4
with
V
4
=
{
a,b,c,d,e,f
}
, E
4
=
{
ab,bc,cd,de,ae,ef,cf
}
Solution.
Isomorphic (two isomorphisms) with the vertex correspondences:
1
f,
2
c,
3
d,
4
e,
5
a,
6
b
or
1
f,
2
e,
3
d,
4
c,
5
b,
6
a
(c)
G
5
with
V
5
=
{
a,b,c,d,e,f,g,h
}
, E
5
=
{
ab,bc,cd,ad,ef,fg,gh,he,dh,bf
}
G
6
with
V
6
=
{
s,t,u,v,w,x,y,z
}
, E
6
=
{
st,tu,uv,sv,wx,xy,yz,wz,sw,vz
}
Solution.
Not isomorphic: they have the same number of vertices, edges, and set of vertex de-
grees. But the degree 2 vertices of
G
1
are all adjacent to two degree 3 vertices, while the degree 2
vertices of
G
2
are all adjacent to one degree 2 vertex and one degree 3 vertex.
Problem
2.
Deﬁnition
??
. A graph is
connected
iff there is a path between every pair of its vertices.
False
Claim.
If every vertex in a graph has positive degree, then the graph is connected.
(a)
Prove that this Claim is indeed false by providing a counterexample.
Creative Commons
2010,
Prof. Albert R. Meyer
.