364
(b) The planar graph shown at the right, which is isomorphic to the
graph
of
the icosahedron (see Exercise
13,
Section
10.2),
has
twelve vertices, each
of
degree
5.
Solutions to Exercises
19.
(a) [BB] Let
91,92,
...
,9n
be the connected components
of
9.
Since
9i
has at least three vertices,
we have
Ei
:::;
3Vi

6. Hence,
L:
Ei
:::;
3
L:
Vi

6n,
so
E
:::;
3V

6n
as required.
(b)
Let
91, 92,
...
,9n
be the connected components
of
9.
If
9i
has at least three vertices, then
Ei
:::;
3Vi

6.
If
9i
has two vertices, then
Ei
=
1,
so
Ei
=
3Vi

5.
If
9i
has one vertex, then
Ei
=
0 and
Ei
=
3Vi

3.
In all cases,
Ei
:::;
3Vi

3,
so
E
=
L:
Ei
:::;
3
L:
Vi

3n
=
3V

3n.
Finally, note that
if
n
~
2,
then
3V

3n
:::;
3n

6
so Theorem
13.1.4
holds while,
if
n
=
1,
the
graph is connected; we established
E
:::;
3V

6 for such a graph in the text.
20.
We
may assume in all parts
of
this question that
9
is connected.
(a) [BB] Say there is only one vertex
of
degree at most
5.
Then
L: deg
Vi
~
6(V

1)
=
6V

6,
contradicting
L: deg
Vi
=
2E
:::;
6V

12.
(b)
Say there are only two vertices
of
degree at most
5.
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 Summer '10
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 Graph Theory, Planar graph, 3 L, deg

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