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Section 10.2
273
14. The reader will appreciate that it is difficult to insert threedimensional models within the covers of a
book!
15. (a) [BB] As suggested, add an extra vertex
v
to
g
and join it to all other vertices. Then deg
v
=
n
~
nt
l
,
and deg
w
~
n2l
+
1
=
~
for all other vertices. By Theorem 10.2.4, this new graph with
n
+
1 vertices has a Hamiltonian cycle. Deleting
v
and all the new edges incident with
v
leads to
a Hamiltonian path in our original graph.
(b) The first graph satisfies the condition in (a), so it has a Hamiltonian path. One is shown to the left
below.
1
2
(c) The second graph has a Hamiltonian path as shown on the right above.
(d) No. The graph on the right above is a counterexample since it has a Hamiltonian path while
n2l
=
~
=
4!
and not all vertices have degree at least 5. Another example is the graph
00
000 .
This certainly has a Hamiltonian path, but n2l
=
5
2
1
=
2 and the end vertices are
of degree one.
Also, the converse to Dirac's Theorem is false, as we can see by the graph on the right above. This
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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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