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
can
be
continued without repeating an arc.
It
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 Summer '10
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 Graph Theory, Polyhedron, vertices, adjacency matrix, Eulerian Trail, arc yw

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