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
.
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.
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.