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368
Solutions to Exercises
1
2~2
\il5JI
1
3
(b) [BB] The converse of the FourColor Theorem states that if the chromatic number of a graph
is at most four, then the graph is planar. This result is false. The Petersen graph is not planar
(Exercise 4 of Section 13.1), but, as we saw in part (a),
X
=
3.
7. (a) [BB] Yes, a tree is planar and we can prove this by induction on
n,
the number of vertices. Cer
tainly a tree with one vertex is planar. Then, given a tree with
n
vertices, removing a vertex of
degree
1
(and the edge with which it is incident) leaves a tree with
n
1
vertices which is planar by
the induction hypothesis. The deleted vertex and edge can now be reinserted without destroying
planarity.
(b) Since a tree is connected and planar, we must have
V

E
+
R
=
2. But
R
=
1, hence,
V

E
=
1
or
E
=
VI,
as desired.
(c) The proofthat
X(T)
=
2 is perhaps easiest by induction on
n,
the number of vertices.
If
n
=
2,
the tree must be
00
and this can be colored with 2 colors, but not with fewer. Assuming the
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This note was uploaded on 11/08/2010 for the course MATH discrete m taught by Professor Any during the Summer '10 term at FSU.
 Summer '10
 any
 Graph Theory

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