Section 10.2
275
11.
If
i
>
j
>
3, then
VI V2Vjl Vj
is a 4cycle.
111.
If
i =
4 and
j
=
3, then
VI
is adjacent to exactly half of
V5,
..•
,Vn
while
V2
is adjacent to the
other half.
If
V2
is adjacent to
V5,
choosing
k
>
5 minimal with
V2
not adjacent to
Vk,
we have the
4cycle
VI V2Vkl Vk.
If
VI
is adjacent to
V5,
choose
k
> 5 minimal with
VI
and
Vk
not adjacent.
Then
VI V2VkVkl
is a 4cycle.
iv.
If
j
=
n
and
VI
is adjacent to
Vn
l,
then
VI V2Vn Vn
l
is a 4cycle.
v.
If
j
=
n
and
V2
is adjacent to
Vnlo
then
VI V2Vnl Vn
is a 4cycle.
We note that the case
j
=
n,
i =
n
 1 is not possible.
VI.
If
n
>
j
>
i
and
VI
is adjacent to
Vj+l,
then
VI V2VjVj+!
is a 4cycle.
vii.
If
n
>
j
>
i
and
V2
is adjacent to
Vj+!,
then
VI V2 Vj+
1
Vj
is a 4cycle.
21. (a) False. A Hamiltonian cycle in a graph contains no proper cycles. The graph itself, on the other
hand, can be expected to contain many proper cycles, for example, any complete graph
Kn
(n
>
3).
(b) The complete graph
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 Summer '10
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 Graph Theory, Glossary of graph theory, hamiltonian cycle, Graph theory objects, Eulerian path, Hamiltonian path, Eulerian

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