262
Solutions to Review Exercises
Proof. (
f )
If
n
=
2m,
the complete bipartite graph
Kd,d
has the required property.
(
f )
Now assume that
9
is a bipartite graph with
n
vertices each of degree
d.
Label the bipartition
sets VI and
V2,
and suppose they contain
nl and
n2
vertices, respectively. Now count the edges in
9
and note that each edge contributes 1 to the sum of the degrees of the vertices in VI and one to the sum
of the degrees of the vertices in
V2.
Thus, we discover
dnl
=
L
degv
=
L
degv
=
dn2,
VEVl
VEV2
so
nl
=
n2
and
n
=
nl
+
n2
is even. Now a vertex
v
in VI is adjacent only to vertices in
V2,
so
deg
v
=
d
::;
n2
=
~.
This completes the argument.
10. (a) Assume this is possible. Then two of the vertices of
1£
would have to belong to the same bipartition
set in
9
(since
1£
has at least three vertices).
In
a bipartite graph, however, vertices in the same
bipartition set are not adjacent. This contradicts the fact that
1£
is complete, so no such situation
is possible.
(b) This is indeed possible. For example, let
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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, Sets

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