Open Journal of Discrete Mathematics
Vol.3 No.2(2013), Article ID:30170,8 pages DOI:10.4236/ojdm.2013.32015
Longest Hamiltonian in Nodd-Gon
Depto de Matemática, Universidad Nacional del Sur, Bahía Blanca, Argentina
Email: biniel@criba.edu.ar
Copyright © 2013 Blanca I. Niel. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Received February 4, 2013; revised March 26, 2013; accepted April 15, 2013
Keywords: Hamiltonian path; Extremal problems; Euclidean geometric problem; Farthest Neighbor tours; Traveling Salesman Problem; Geometry of Odd Regular Polygons
ABSTRACT
We single out the polygonal paths of order that solve each of the
different longest non-cyclic Euclidean Hamiltonian path problems in
networks by an arithmetic algorithm. As by product, the procedure determines the winding index of cyclic Hamiltonian polygonals on the vertices of a regular polygon.
1. Introduction
Our aim implies to determine the overall lengths of every Longest Euclidean Hamiltonian Path Problems and the composition and the orderings of the directed segments that accomplish these longest Hamiltonian travels. The identification regardless of planar rotation and orientation is done with the proposed algorithm [1-3].
This paper apart from the Introduction, Conclusion and References contains §2 Algorithm and Hamiltonian Paths in Nodd-Gons and §3 Maximum Hamiltonian Path Problems in Nodd-Gons. §2 formulates specific Max. Hamiltonian Problems and postulates the algorithm for their resolutions. §3 devoted to the solution of the
different Max. Traveling Salesman Path Problems in Nodd-Gons [4,5].
2. Algorithm and Hamiltonian Paths in Nodd-Gons
This work is focused in the resolution of the
different Maximum Traveling Salesman Path Problems of order with inicial point at
and final point at
for
(see Figure 1) in the
networks. These structures are built by the complete graph
on the odd regular polygon vertices, i.e.
, and weighted with the Euclidean distances
between nodes [6].
2.1. Intrinsic Geometry and Arithmetic
Let be the points of the
set and let them be clockwise enumerated by the integers modulo
,
, from the vertex
. For each
in
and each
, let
denote the segment that joins
with
, while
denotes the one that joins
to
. From now onwards,
and
denote to
and
, respectively. Let
be the diameter, it joins the vertex
with its opposite
, only if
is even.
and
respectively designate the quasi-diameters if
is odd (see Figure 1), [7].
If symbolizes a regular n-gon inscribed in the unitary circle and with vertices in
,
can be considered as the polygonal of sides
[8]. From the vectorial interpretation of the
segments,
can be interpreted as the resultant of the polygonal of
sides of
, that joins clockwise
to
, while
is the resultant of the polygonal of
sides that joins clockwise
to
.
Figure 1. Lk segments in Nodd-Gons.
The segments and
are the respective chords (or resultants) of the polygonals
and
consecutive sides of
, whichever are the integers
and
. Therefore, it is natural to associate
with the integer
, and likewise
with the integer
.
Definition 2.1.1 For any integer,
is a
segment if for any
,
, and for any
,
is equal to
or
.
Definition 2.1.2 If is an
segment, the integer associated to
, noted as
, is given by:
Definition 2.1.3 If is a sequence of
segments, the integer associated to the path evolved by
, is given by
.
It should be taken into account the following facts:
• The consecutive collocation of two segments from any vertex
determines the vertex that corresponds to collocate, from
and in clockwise, as many sides of
as correspond to the sum of the integers associated to each one of the two
segments. In other words, the resultant of a polygonal built by two
segments, is other
segment and its associated integer is the sum (modulo
) of the integers associated to the components of the polygonal.
• The segment is
by considering any fixed value of
, when
. Otherwise, if
, is
.
The concept of the associated integer and its addition modulo
, deploy the following geometric correlate over the set of vertices
: For each
,
, the geometric place that corresponds to the vertex
coincides with the place that corresponds to
, for each integer
. Since the segments
and
respectively connect the
vertices to
and
to
, it is clear that for any integer
between 0 and
, the vertices
and
are symmetric with respect to the horizontal axis. Given a sequence of
segments, henceforward the polygonal that they determine is in a oneto-one relationship with the sum of each one of these directed segments that belong to the sequence.
Since, whichever
and
are, without loss of generality in the sequences of
segments, the second subindices of these directed segments are rooted out.
2.2. Resuming the Algorithm
Lemma 2.6 and Theorem 2.7 in [1] detail the proofs of the following algorithmic statements.
Theorem 2.2.1 The pathway determined by a sequence of
segments starts and ends at the same vertex
if and only if
.
Theorem 2.2.2 A sequence of
segments determines a Euclidean Hamiltonian cycle
of order
if and only if any proper subsequence of order
has associated integer neither
nor a multiple of
and
.
Corollary 2.2.1 A sequence of
segments of order
,
, building a Euclidean closed polygonal in
networks, passing once through certain or all
vertices, has
.
Since, is a multiple of
exists
less than or equal to
which counts the times that
cw.
winds around the geometric centre of. We named this specific integer as the “winding index”.
2.3. Applications of the Algorithm: Winding Index in Special Cyclic Paths in Nodd-Gons
Let symbolize a cyclic polygonal in
network, which does not repeat vertices, with the exception of the first and the last one, and which passes through certain
nodes,
. Specially,
stands for Euclidean Hamiltonian cycles in
network.
Exampe 2.1. Let
. If
does not divide
they are
s of winding index
[9].:
is the Max TSP [10].
Exampe 2.2. Let
.
The angular cw. avance is proportional to:
then is the winding index. Algorithmic computations render that these cycles are
and
for networks built on
. For
the algorithm prompts
as winding index and singled out them as and
if
. In
,
, the algorithm characterizes these cycles as
in
with winding index
.
Exampe 2.3. Table 1 deploys cycles living in
.
Exampe 2.4. Table 2 shows Euclidean Hamiltonian cycles in special networks.
3. Maximum Hamiltonian Path Problems in Nodd-Gons
In network for
we study the trajectories built by a single
segmentTable 1.
in
.
Table 2. in
.
directed
segments, and
directed segments, that is (1).
3.1. Lengths of Relevant Pathways
Our present concern is to study the Euclidean lengths and the composition of the directed segments that build the trajectories given by (1).
(1)
Since for the lengths
of the segments
,
verify the following relationships:
Therefore, the overall traveled Euclidean lengths of the pathways (1) are given by:
(2)
Therein, precisely we focusing on the Euclidean Hamiltonian cycles, s, which accomplish the lengths
(2) in network.
Next Theorem establishes the composition of the directed segments that give birth to the sequences with overall traveled lengths (2).
Theorem 3.1.1 The overall traveled lengths (2) in
are accomplished for any sequence built by a single
,
,
,
and
directed segments if
and
if the conditions in (3) are satisfied.
(3)
Proof
From the constraints and
follows
should be
and hence
. Therefore, the admissible couples
for the lengths (2) should verified (3). □
Backward recurrence over the traveled length in steepest descent steps from the max to
constraint and the lack of Hamiltonian cycles for
state that (4) is the Euclidean Hamiltonian Maximum Path length when is rooted out.
(4)
3.2. Specific Directed Segments for the Max. Traveling Salesman Path Problems in Nodd-Gons
We confirm in Theorem (3.3.1), Theorem (3.3.2) and Theorem (3.3.3) the existence of Euclidean Hamiltonian cycles that attain the overall Euclidean lengths given by the sequences (1) and the assignments (3).
1) For if
and
in
(3) exists s with overall traveled length (2). See Theorem (3.3.1) at pg. 4.
2) For
a) in (3) exists
s with whole traveled length (2). See Theorem (3.3.2) at pg. 5b)
and
in (3) exists
s with whole traveled length (2). See Theorem (3.3.3) at pg. 5.
3.3. Orderings of the Directed Segments for the Max. Traveling Salesman Paths in Nodd-Gons
symbolizes any Euclidean Hamiltonian path that resolves the Max Traveling Salesman Path Problems with initial vertex
and final vertex
, that is whichever be the bridge,
for
between the starting and ending points.
Observation 3.1 Proofs of Theorem 3.3.1, Theorem 3.3.2 and Theorem 3.3.3 result from direct application of Theorem 2.2.2 of the proposed algorithm.
Theorem 3.3.1 Let an odd integer for
. The pathways (5) and (6) build
s in
networks if
.
(5)
(6)
for. □
Let and
denote, respectively, the forward and backward readings of any sequence of
segments.
Corollary 3.3.1 In
networks if, forward and backward readings of the sequences (5) and (6) are
.
Consequently, and
of the sequence (5) and
(6) account for 2 plus to distinct sequencesrespectively. Furthermore,
and
of the pathway (5) and paths (6) build
s if the directed segment
is initially appended to these sequences. □
Theorem 3.3.2 Let an even integer for
. The pathways (7) and (8) build
s in
networks if
, with
is the number of
and
, respectively.
(7)
(8)
for □
Corollary 3.3.2 In networks if
, forward and backward readings of the sequences (7) and (8) are
. Particularly, the enumeration of the distinct
s given birth from the forward and backward readings of the sequences (8) depend on the
evenness. Specifically1) If
is odd, since
every sequence in (8) is not a palindrome [1]. Moreoverthe sequences defined in (8) are in couples
and
Specifically, the
path
determined by
coincides to
path
determined by
,
path coincides with
of the sequence defined by
and so on. That is the
paths defined by (8) with
coincide with the
paths determined by (8) with
.
Therefore, exists distinct
s which correspond with each one of the
determined by (8). Since
of (7) is different to its
, both
s should be added to the final enumeration. In conclusion, the distinct
s are
.
2) If is even, since
, then
this index in (8) builds a
which is a palindrome [1]. In addition,
paths defined by (8) with coincide with the
paths determined by (8) with
. Therefore, exists
distinct
s which correspond with each one of
paths determined by (8). Since
of (7) is different to its, both
s should be added to the final enumeration. In conclusion, the distinct
s are
. □
Theorem 3.3.3 Let an even integer for
. The pathways (9) build
s in
networks if
, meanwhile
is the number of
and
the amount of
.
(9)
for. □
Corollary 3.3.3 In networks if
, forward and backward readings of the sequences (9) are
s. Particularly, the enumeration of the distinct
s given birth from the forward and backward readings of the sequences (9)
depend on the evenness. Specifically,
1) If is odd, i.e.
is even, then
, therefore the sequence in (9) build by this index
is a palindrome [1]. Moreover,
sequences defined in (9) are in couples
and
with the exception of that given birth by the index
which its
and
is exactly the same pathway at all.
Specifically, the path
determined by
coincides to
path
determined by
,
path coincides with
of the sequence defined by
and so on, until the index
at which
and beget only one path. That is the
paths defined by (9) with the downgraded indexes
coincide with the
paths determined by (9) with
.
In conclusion, exists distinct
s which correspond with each one of the
path determined by (9).
2) If is even, i.e.
is odd, since
, then sequences (9) build
s none of them are palindrome [1]. In addition,
paths of the indexes coincides with
paths of the downgraded indexes
, respectively. In conclusion, exists
distinct
s vis-à-vis with each one of the
path determined by (9). □
Observation 3.2 Corollary 3.3.1, Corollary 3.3.2 and Corollary 3.3.3 result from Theorem 3.3.1, Theorem 3.3.2 and Theorem 3.3.3, respectively.
In conclusion, the s which resolve the Max.
Euclidean Hamiltonian Path Problems with the
as the bridge between the endings of the Hamiltonian paths are evolved by the sequences (5) and (6) if. Otherwise by the orderings (7)-(9). Moreover, with the exception of the palindromes their backward readings also resolve these specific Max. Traveling Salesman Problems.
3.4. Bicoupled Nodd-Gons TSP Conjeture
We choose the geometric paths that start up at of the quasi-spherical mirror of unitary radius, touch
times-including the last touchinganywhere on the hollowed mirror, and end up at
, with
In this geometry each
array of angles
, see Figure 2, denoted
, determines a path with
verticesincluding the initial and arrival pointsand
linear branches, [8,11,12]. This path may have two or more coincident vertices and linear branches shrunk to a point. For each
the
angles
are selected (see Figure 2) as independent variables of the overall traveled length function of the paths
.
The length of the geometric path determined by, is given by (10)
(10)
When,
, for any polygonal cyclic trajectory, there is an
-array
which characterizes them. In particular, amongst these pathways are those that have as vertices the
points, with
See [10] Theorem 2.1.1. and Appendix A, from page 78 to 80 [8]. Let
(11)
be a generalized length of (10), where are the analogous angular parameters with the restraints
and
, and
in
is the structural parameter for the locations of the coupled vertices of the inner
-polygon,
networks [3].
Herein, see Figure 3, we pose the following conjeture: Are Max. TSPs in bilayer
networks baited for the regular shapes of the Max. TSP in
networks?
4. Conclusion
This paper is an offspring of a series of previous works about Hamiltonian maximum overall traveled lengths in
networks. Herein are singled out all the Euclidean Hamiltonian pathways that resolve
Figure 2. Measure of αi angular parameter.
Figure 3. Max TSP in coupled Nodd-Gons.
the different maximum traveled Hamiltonian paths of order
in
networks. As a by-product the proposed algorithm allow us to determine the winding index of specific cyclic polygonals. The approach is a step forward on the intrinsic geometry and inherent arithmetic of the vertex locus of the Nodd-Gons.
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