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Counting cycles in planar triangulations
Authors
On-Hei Solomon Lo
Carol T. Zamfirescu
Publication date
3 October 2022
Publisher
View
on
arXiv
Abstract
We investigate the minimum number of cycles of specified lengths in planar
n
n
n
-vertex triangulations
G
G
G
. It is proven that this number is
Ξ©
(
n
)
\Omega(n)
Ξ©
(
n
)
for any cycle length at most
3
+
max
β‘
{
r
a
d
(
G
β
)
,
β
(
n
β
3
2
)
log
β‘
3
2
β
}
3 + \max \{ {\rm rad}(G^*), \lceil (\frac{n-3}{2})^{\log_32} \rceil \}
3
+
max
{
rad
(
G
β
)
,
β(
2
n
β
3
β
)
l
o
g
3
β
2
β}
, where
r
a
d
(
G
β
)
{\rm rad}(G^*)
rad
(
G
β
)
denotes the radius of the triangulation's dual, which is at least logarithmic but can be linear in the order of the triangulation. We also show that there exist planar hamiltonian
n
n
n
-vertex triangulations containing
O
(
n
)
O(n)
O
(
n
)
many
k
k
k
-cycles for any
k
β
{
β
n
β
n
5
β
,
β¦
,
n
}
k \in \{ \lceil n - \sqrt[5]{n} \rceil, \ldots, n \}
k
β
{β
n
β
5
n
β
β
,
β¦
,
n
}
. Furthermore, we prove that planar 4-connected
n
n
n
-vertex triangulations contain
Ξ©
(
n
)
\Omega(n)
Ξ©
(
n
)
many
k
k
k
-cycles for every
k
β
{
3
,
β¦
,
n
}
k \in \{ 3, \ldots, n \}
k
β
{
3
,
β¦
,
n
}
, and that, under certain additional conditions, they contain
Ξ©
(
n
2
)
\Omega(n^2)
Ξ©
(
n
2
)
k
k
k
-cycles for many values of
k
k
k
, including
n
n
n
Similar works
Full text
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oai:arXiv.org:2210.01190
Last time updated on 22/11/2022