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Section 13.2
369
10. (a)
[BB]
~
(b)(BB]
~
(c) [BB] The graph isn't planar, by Kuratowski's Theorem, as the results of (a) and (b) each illustrate.
(d) [BB] A 3coloring is shown. Since the graph
contains triangles, fewer than three colors
will not suffice. The chromatic number is
3~3
three.
2
1
(e) [BB] The converse of the FourColor Theorem says that a graph with
X
~
4 is planar. This is not
true, as this graph illustrates: The chromatic number is 3, but the graph is not planar.
11. (a) [BB] By PAUSE 7 of this section, for any
n, X(Kn)
=
n
and for any m,
n, X(Km,n)
=
2. Thus,
X(K
14)
=
14 and X(K
5,14)
=
2.
(b) A cycle with 38 edges is a cycle with an odd number (39) of vertices. We must start with two
different colors
R
and
W.
If
we try to alternate these, the odd vertices
R
and the even
W,
say,
then we run into trouble because vertex 39 is adjacent to an even and an odd number vertex (38
and
1).
Hence,
X(91)
=
3. On the other hand,
92
is a cycle with an even number of vertices, so
X(2)
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 Summer '10
 any
 Graph Theory, Angles

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