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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
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 Summer '10
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 Graph Theory

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