354
Solutions to Exercises
7. (a) [BB] Choose a vertex
v.
For each vertex
u
adjacent to
v,
orient the edge
uv
in the direction
u
+
v.
(b) The answer is yes. Since 9 is an Eulerian graph, there exists an Eulerian circuit. Now just orient
the edges of this circuit in the direction of a walk along it.
8. [BB] Yes. To get from
a
to
b
in the new orientation, just find the path from
b
to
a
in the old orientation
and follow it in reverse.
9. (a) [BB] False. The graph shown at the right cannot be given a
strongly connected orientation, yet every edge is part of a
circuit.
(b) True.
If
u
has degree 1 and
u
is incident with the edge e, then orienting e away from
u
makes it
impossible to get to
u
from another vertex and orienting e towards
u
makes it impossible to leave
u.
Either of these situations contradicts the fact that the graph is strongly connected.
(c) True. By Theorem 10.1.4, a connected graph with no odd vertices is Eulerian. Orienting the edges
along an Eulerian circuit in the direction of the circuit produces a strongly connected orientation.
10. (a) [BB]
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 Summer '10
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 Graph Theory, Planar graph, Vertextransitive graph, Eulerian

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