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300
Solutions to Exercises
10. Copy the proof of Theorem 10.1.4. The only change is that we have to be sure it is possible to follow
an arc in the proper direction.
11. [BB] A digraph has an Eulerian trail between vertices
u
and
v
if and only if
• one of these vertices, say
u,
has outdegree one more than its indegree, the other vertex,
v,
has
indegree one more than its outdegree,
• the indegrees and outdegrees of every vertex except
u
and
v
are equal, and
• for every pair of vertices
x, y
there is a (directed) path from
x
to
y
or from
y
to
x.
Proof.
If
there is an Eulerian trail from
u
to
v,
it is
clear that the three given conditions are necessary. On
the other hand, suppose the three conditions hold in a
digraph
9.
We show that
9
has an Eulerian trail. The
result is obvious if
u
and
v
are the only vertices of
9,
so we may assume 9 has at least three vertices.
x
_.
.
.....
T
v
By hypothesis, there is an arc
ux.
If
x
=f.
v,
there is an arc
xy,
and so on. By hypothesis, this process
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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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