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2020 - 2001(3)

Any replacements are listed farther down

[172] **viXra:2001.0462 [pdf]**
*submitted on 2020-01-22 07:38:39*

**Authors:** Theophilus Agama

**Comments:** 8 Pages.

In this paper we study the distribution of boundary points of expansion. As an application, we say something about the lonely runner problem. We show that given $k$ runners $\mathcal{S}_i$ round a unit circular track with the condition that at some time $||\mathcal{S}_i-\mathcal{S}_{i+1}||=||\mathcal{S}_{i+1}-\mathcal{S}_{i+2}||$ for all $i=1,2\ldots,k-2$, then at that time we have \begin{align}||\mathcal{S}_{i+1}-\mathcal{S}_i||>\frac{\mathcal{D}(n)\pi}{k-1}\nonumber
\end{align}for all $i=1,\ldots, k-1$ and where $\mathcal{D}(n)>0$ is a constant depending on the degree of a certain polynomial of degree $n$. In particular, we show that given at most eight $\mathcal{S}_i$~($i=1,2,\ldots, 8$) runners running round a unit circular track with distinct constant speed and the additional condition $||\mathcal{S}_i-\mathcal{S}_{i+1}||=||\mathcal{S}_{i+1}-\mathcal{S}_{i+2}||$ for all $1\leq i\leq 6$ at some time $s>1$, then at that time their mutual distance must satisfy the lower bound\begin{align}||\mathcal{S}_{i}-\mathcal{S}_{i+1}||>\frac{\pi}{7C\sqrt{3}}\nonumber
\end{align}for some constant $C>0$ for all $1\leq i \leq 7$.

**Category:** Combinatorics and Graph Theory

[171] **viXra:2001.0437 [pdf]**
*submitted on 2020-01-21 16:42:34*

**Authors:** N. Durga, S. Satham Hussain, Saeid Jafari, Said Broumi

**Comments:** 26 Pages.

In this manuscript, the operations on neutrosophic vague graphs are introduced. Moreover, Cartesian product, cross product, lexicographic product, strong product and composition of neutrosophic vague graph are investigated and the proposed concepts are illustrated with examples.

**Category:** Combinatorics and Graph Theory

[170] **viXra:2001.0404 [pdf]**
*submitted on 2020-01-20 18:17:28*

**Authors:** Theophilus Agama

**Comments:** 6 Pages.

We introduce and study the needle function. We prove that this function is a function modeling $n$-step self avoiding walk. We show that the total length of the $l$-step self-avoiding walk modeled by this function is of the order \begin{align}\ll \frac{n^{\frac{3}{2}}}{2}\bigg(\mathrm{\max}\{\mathrm{sup}(x_j)\}_{1\leq j\leq \frac{l}{2}}+\mathrm{max}\{\mathrm{sup}(a_j)\}_{1\leq j\leq \frac{l}{2}}\bigg).\nonumber
\end{align}

**Category:** Combinatorics and Graph Theory

[169] **viXra:1911.0451 [pdf]**
*submitted on 2019-11-26 13:49:43*

**Authors:** M. D. Sheppeard

**Comments:** 4 Pages.

In this lecture we count the number of integer partitions P(n) using an elementary algorithm based on the combinatorics of trees, coded using free apps on the iPad.

**Category:** Combinatorics and Graph Theory

[168] **viXra:1911.0203 [pdf]**
*submitted on 2019-11-11 00:47:10*

**Authors:** A. A. Frempong

**Comments:** 38 Pages. Copyright © by A. A. Frempong

The solution of the traveling salesman problem (TSP) in this paper makes this problem no longer an NP-hard problem, but rather, a P problem. The TSP was solved in polynomial time and its solution was also correctly checked in polynomial time. Also solved were an NP-Complete TSP, and six other NP-Complete problems. The TSP solution killed two (three) birds with one stone, because its solution made the NP-hard problems and NP-Complete problems become P problems. The shortest route as well as the longest route for the salesman to visit each of nine cities once and return to the base city was determined. In finding the shortest route, the first step was to arrange the data of the problem in increasing order, since one's interest is in the shortest distances; but in finding the longest route, the first step was to arrange the data of the problem in decreasing order, since one's interest is in the longest distances. For the shortest route, the main principle is that the shortest route is the sum of the shortest distances such that the salesman visits each city once and returns to the starting city; but for the longest route, the main principle is that the longest route is the sum of the longest distances such that the salesman visits each city once and returns to the starting city. Since ten cities are involved, ten distances would be needed for the salesman to visit each of nine cities once and return to the starting city. One started the construction of the shortest route using only the shortest ten distances; and if a needed distance was not among the set of the shortest ten distances, one would consider distances longer than those in the set of the shortest ten distances. For the longest route, the construction began using only the longest ten distances; and if a needed distance was not among the set of the longest ten distances, one would consider distances shorter than those in the set of the longest ten distances. It was found out that even though, the length of the shortest or the longest route is unique, the sequence of the cities involved is not unique. The approach used in this paper can be applied in work-force project management and hiring, as well as in a country's work-force needs and immigration quota determination. Since approaches that solve the TSP and NP-Complete problems can also solve other NP problems, and the TSP and NP-Complete problems have been solved, all NP problems can be solved. If all NP problems can be solved, then all NP problems are P problems, and therefore, P is equal to NP. The CMI Millennium Prize requirements have been satisfied.

**Category:** Combinatorics and Graph Theory

[167] **viXra:1911.0056 [pdf]**
*submitted on 2019-11-04 08:50:15*

**Authors:** Colin James III

**Comments:** 1 Page. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

We evaluate the chromatic number of the product of two ℵ1-chromatic graphs as countable. It is not tautologous as are then also the derived consistent and relatively consistent conjectures for k=3. These results form a non tautologous fragment of the universal logic VŁ4.

**Category:** Combinatorics and Graph Theory

[166] **viXra:1910.0283 [pdf]**
*submitted on 2019-10-16 23:07:31*

**Authors:** Yasushi Ieno

**Comments:** 15 Pages.

Given two sets, one consisting of variables representing distinct positive n numbers, the other set `a kind of power set' of this n-element set. I got interested in the fact that for the latter set, depending on the values of two elements, it can occur that not every pair of elements is `comparable', that is to say, it is not always uniquely determined which of two elements is smaller. By proving theorems in order to go ahead with our research, we show a table which describes for how many `comparable' cases exist, for several n's.

**Category:** Combinatorics and Graph Theory

[165] **viXra:1910.0230 [pdf]**
*submitted on 2019-10-14 05:00:40*

**Authors:** Yasushi Ieno

**Comments:** 7 Pages.

The 6th problem of the 50th International Mathematical Olympiad (IMO),
held in Germany, 2009, is called 'the grasshopper problem'. To this problem
Kos developed theory from unique viewpoints by reference of Noga Alon's
combinatorial Nullstellensatz.
We have tried to solve this problem by an original method inspired by a
polynomial that Kos defined, then researched for n=3 and n=4. For
almost cases the problem can be solved, but there remains imperfection due
to 'singularity'.

**Category:** Combinatorics and Graph Theory

[164] **viXra:1909.0632 [pdf]**
*submitted on 2019-09-30 14:19:34*

**Authors:** Colin James III

**Comments:** Pages.

We evaluate the definition of Petri nets from F ⊆ (P × T) ∪ (T × P) as F ⊆ (P → T) ∪ (T → P) where F, P, T stand for flow, places, tokens. It is not tautologous. This further disallows the conjecture of verification of work flows by reachability. These conjectures form a non tautologous fragment of the universal logic VŁ4.

**Category:** Combinatorics and Graph Theory

[163] **viXra:1909.0066 [pdf]**
*submitted on 2019-09-03 12:51:30*

**Authors:** Ortho Flint, Stuart Rankin

**Comments:** 10 Pages. The labelled cycle-decomposition trees are a powerful invariant and computationally inexpensive to produce.

The notion of a labelled cycle-decomposition tree for an arbitrary graph is introduced
in this paper.
The idea behind the labelled cycle-decomposition tree, one constructed for
each vertex in the graph, is to attach to each vertex a data structure
that gives more than local information about the vertex, where the data
structures can be compared in polynomial time. The fact that trees can be compared
in linear time, with particularly efficient algorithms for
solving the isomorphism problem for rooted and labelled trees, led us to
consider how we could represent the vertex's view of the graph by means of
a labelled rooted tree. Of course, cycles in the graph can't be directly
recorded by means of a tree structure, but in this article, we
present one method for recording certain of the cycles that are encountered
during a breadth-first search from the vertex in question. The collection of
labelled, so-called cycle-decomposition trees for the graph, one for each
vertex, provide an invariant of the graph.

**Category:** Combinatorics and Graph Theory

[162] **viXra:1908.0480 [pdf]**
*submitted on 2019-08-24 01:19:30*

**Authors:** A. A. Frempong

**Comments:** 10 Pages. Copyright © by A. A. Frempong

By applying differential and integral calculus, this paper covers the principles and procedures for producing the solution of a problem, given the procedure for checking the correctness of the solution of a problem, and vice versa. If one is able to check quickly and completely, the correctness of the solution of a problem, one should also be able to produce the solution of the problem by reversing the order of the steps of the checking process, while using opposite operations in each step. The above principles were applied to four examples from calculus as well as to an example from geometry. Even though in calculus, one normally uses differentiation to check the correctness of an integration result, one will differentiate a function first, and then integrate the derivative to obtain the original function. One will differentiate the trigonometric functions, tan x, cot x, sec x and csc x; followed by integrating each derivative to obtain each original function. The results show that the solution process and the checking process are inverses of each other. In checking the correctness of the solution of a problem, one should produce the complete checking procedure which includes the beginning, the middle, and the end of the problem. Checking only the correctness of the final answer or statement is incomplete checking. To facilitate complete checking, the question should always be posed such that one is compelled to show a complete checking procedure from which the solution procedure can be produced. A general application of P = NP is that, if the correctness of the solution of a problem can be checked quickly and it is difficult to write a solution procedure, then first, write a complete checking procedure and reverse the order of the steps while using opposite operations to obtain the solution of the problem. Therefore, P is equal to NP.

**Category:** Combinatorics and Graph Theory

[161] **viXra:1907.0362 [pdf]**
*submitted on 2019-07-18 12:55:28*

**Authors:** A. Sugumaran, V. Mohan

**Comments:** 13 Pages.

Abstract. A difference cordial labeling of a graph G is a bijective function f from V(G) onto {1, 2, 3, ⋯ , |V(G)|} such that each edge uv is assigned the label 1 if |f(u) – f(v)| = 1, and the label 0 otherwise, satisfying the condition that the number of edges labeled with 1 and the number of edges labeled with 0 differ by at most 1. A graph with difference cordial labeling is called a difference cordial graph. In this paper we proved that the umbrella graph U(m, n), duplication of a vertex by an edge in a cycle Cn, duplication of an edge by a vertex in a cycle Cn and the total graph of a path Pn are difference cordial graphs.

**Category:** Combinatorics and Graph Theory

[160] **viXra:1907.0328 [pdf]**
*submitted on 2019-07-16 06:58:12*

**Authors:** John Archie Gillis

**Comments:** 30 Pages.

The present article takes a novel approach to solving NP-Complete problems and provides steps that a computational device and software program can follow to accurately solve NP-class problems without the use of heuristics or brute force methods. The present methods are fast and accurate if utilized properly.
The paper states that the solution to solving the P vs NP question (and our ability to design algorithms to solve such problems efficiently) lies in a novel method presented for searching, filtering, combining and structuring data, which describes a novel method for breaking specific problems into logical groupings that the present inventor (John Archie Gillis) has defined as collaborative variables. They utilize novel binary representations/conversions, so that one can more easily and quickly determine selected and desired informational outputs.
Many have stated that a solution to this problem will create the world’s first trillionaire as it addresses many pattern-matching and optimization problems that are of great practical interest, such as determining the optimal arrangement of transistors on a silicon chip, developing accurate financial-forecasting models, or analyzing protein-folding behaviour in a cell. Since all the NP-complete optimization problems become easy with the present methods, everything will be much more efficient. Transportation of all forms can now also be scheduled optimally to move people and goods around quicker and cheaper. Manufacturers can improve their production to increase speed and create less waste.
Developments in vision recognition, language comprehension, translation and many other learning tasks will now become much simpler as well. It is felt that by utilizing the systems of the present invention in numerous fields, that the present paper will have profound implications for mathematics, cryptography, algorithm research, artificial intelligence, game theory, internet packet routing, multimedia processing, philosophy, economics and many other fields.

**Category:** Combinatorics and Graph Theory

[159] **viXra:1906.0502 [pdf]**
*submitted on 2019-06-27 07:21:45*

**Authors:** Daria Grushka, Viktoriia Lebid

**Comments:** 4 Pages. Text in Ukrainian. Mohyla Mathematical Journal, Vol 1 (2018) http://mmj.ukma.edu.ua/article/view/152600

Spectral graph theory uses the eigenvalues of matrices associated with a graph to determine the structural properties of the graph. The spectrum of the generalized adjacency matrix is considered in the paper. Graphs with the same spectrum are called cospectral. Is every graph uniquely determined by its spectrum (DS for short)?
This question goes back for about half a century, and originates from chemistry. In 1956 Gunthard and Primas raised the question in a paper that related the theory of graph spectra to Huckel’s theory. At that time it was believed that every graph is determined by the spectrum, until in 1957 Collatz and Sinogowitz presented a pair of cospectral trees. In 1967 Schwenk proved that for almost all trees there is another tree with the same spectrum. Such a statement is neither proved nor refuted for the class of graphs in general. Till now, computational experiments were done on the set of all graphs on up to 12 vertices by Haemers. Computer enumerations for small n show that up to 10 vertices the fraction of graphs that are DS decreases, but for n = 11 and n = 12 it increases again.
We consider the construction of the cospectral graphs called GM-switching for graph G taking the cycle C2n and adjoining a vertex v adjacent to half the vertices of C2n. For these graphs we determine the pairs of cospectral nonisomorphic graphs for small n. It is an operation on graphs that leaves the spectrum of the generalized adjacency matrix invariant. It turns out that for the enumerated cases a large part of all cospectral graphs comes from GM switching, and that the fraction of graphs on n vertices with a cospectral mate starts to decrease at some value of n < 11 (depending on the matrix). Since the fraction of cospectral graphs on n vertices constructible by GM switching tends to 0 if n → ∞, the present data give some indication that possibly almost no graph has a cospectral mate.
Haemers and Spence derived asymptotic lower bounds for the number of graphs with a cospectral mate from GM switching.

**Category:** Combinatorics and Graph Theory

[158] **viXra:1906.0501 [pdf]**
*submitted on 2019-06-27 07:39:47*

**Authors:** Marco Ripà

**Comments:** 12 Pages. Original research paper © Marco Ripà - unpublished and unsubmitted before on any journal

In this paper, we show enhanced upper bounds of the nontrivial n_1 × n_2 × n_3 points problem for every n_1 ≤ n_2 ≤ n_3 < 6. We present new patterns that drastically improve the previously known algorithms for finding minimum-link covering paths, solving completely a few cases (e.g., n_1 = n_2 = 3 and n_3 = 4).

**Category:** Combinatorics and Graph Theory

[157] **viXra:1906.0350 [pdf]**
*submitted on 2019-06-20 03:54:55*

**Authors:** Prajnanaswaroopa S, J Geetha, K Somasundaram

**Comments:** 3 Pages.

In this short note, we give a coloring procedure for graphs which consist of cliques sharing at most one point

**Category:** Combinatorics and Graph Theory

[156] **viXra:1906.0031 [pdf]**
*submitted on 2019-06-03 12:10:14*

**Authors:** Henning Thielemann

**Comments:** 5 Pages. Language: German

The dice sequence
The dice sequence is an adaption of Kruskal's card trick to dice.
We compute precise and approximated probabilities that the trick works.
The adaption to dice simplifies the problem considerably
because the probabilities of the dice rolls are independent.
Initially I wrote the text for Wikipedia but in order to meet Wikipedia's exclusion of original research I wrote up this paper.

**Category:** Combinatorics and Graph Theory

[155] **viXra:1905.0474 [pdf]**
*submitted on 2019-05-23 09:15:36*

**Authors:** Richard J. Mathar

**Comments:** 17 Pages.

This work provides a Java program which constructs free polyominoes of size n sorted by width
and height of the convex hull (i.e., its rectangular bounding box). The results correct counts
for 15-ominoes published in the 1967 proceedings of the SIAM Fall Meeting, and
extend them to 16-ominoes and partially to even larger polyominoes.

**Category:** Combinatorics and Graph Theory

[154] **viXra:1903.0256 [pdf]**
*submitted on 2019-03-13 16:20:16*

**Authors:** Roman Galay

**Comments:** 7 Pages. The following short article offers a couple of algebraically entangled polynomial-time algorithms for the graph isomorphism and clique problems whose correctness is yet to be determined either empirically or through attempting to find proofs.

As it follows from Gödel's incompleteness theorems, any consistent formal system of axioms and rules of inference should imply a true unprovable statement. Actually this fundamental principle can be efficiently applicable in Computational Mathematics and Complexity Theory concerning the computational complexity of problems from the class NP, particularly and especially the NP-complete ones. While there is a wide set of algorithms for these problems that we call heuristic, the correctness or/and complexity of each concrete algorithm (or the probability of its correct and polynomial-time work) on a class of instances is often too difficult to determine, although we may also assume the existence of a variety of algorithms for NP-complete problems that are both correct and polynomial-time on all the instances from a given class (where the given problem remains NP-complete), but whose correctness or/and polynomial-time complexity on the class is impossible to prove as an example for Gödel's theorems. However, supposedly such algorithms should possess a certain complicatedness of processing the input data and treat it in a certain algebraically “entangled” manner. The same algorithmic analysis in fact concerns all the other significant problems and subclasses of NP, such as the graph isomorphism problem and its associated complexity class GI.
The following short article offers a couple of algebraically entangled polynomial-time algorithms for the graph isomorphism and clique problems whose correctness is yet to be determined either empirically or through attempting to find proofs.

**Category:** Combinatorics and Graph Theory

[153] **viXra:1901.0351 [pdf]**
*submitted on 2019-01-24 10:02:02*

**Authors:** Bassey Godwin Bassey

**Comments:** 29 Pages.

In this study we answer questions that have to do with finding out the total number of ways of arranging a finite set of symbols or objects directly under a single line constraint set of finite symbols such that common symbols between the two sets do not repeat on the vertical positions. We go further to study the outcomes when both sets consist of the same symbols and when they consist of different symbols. We identify this form of permutation as 'second-order permutation' and show that it has a corresponding unique factorial which plays a prominent role in most of the results obtained. This has been discovered by examining many practical problems which give rise to the emergence of second-order permutation. Upon examination of these problems, it becomes clear that a directly higher order of permutation exist. Hence this study aims at equipping mathematics scholars, educators and researchers with the necessary background knowledge and framework for incorporating second-order permutation into the field of combinatorial mathematics.

**Category:** Combinatorics and Graph Theory

[152] **viXra:1812.0482 [pdf]**
*submitted on 2018-12-30 19:20:53*

**Authors:** J Gregory Moxness

**Comments:** 4 Pages.

Previously, a determination of the relationship between the Natural numbers (N) and the n'th odd fibbinary number has been made using a relationship with the Golden ratio \phi=(Sqrt[5]+1}/2 and \tau=1/\phi. Specifically, if the n'th odd fibbinary equates to the j'th N, then j=Floor[n(\phi+1) - 1]. This note documents the completion of the relationship for the even fibbinary numbers, such that if the n'th even fibbinary equates to the j'th N, then j=Floor[n(\tau+1) + \tau].

**Category:** Combinatorics and Graph Theory

[151] **viXra:1811.0233 [pdf]**
*submitted on 2018-11-14 06:42:54*

**Authors:** Франц Герман

**Comments:** 3 Pages.

В некоторых задачах, решаемых на компьютере, возникает необходимость нахождения перестановок из n элементов. Для небольших n это можно сделать простым перебором, но с увеличением n, нахождение всех перестановок перебором становится занятием очень трудоёмким. В таком случае удобно применять следующий алгоритм.

**Category:** Combinatorics and Graph Theory

[150] **viXra:1811.0062 [pdf]**
*submitted on 2018-11-04 11:24:53*

**Authors:** Robert DiGregorio

**Comments:** 2 Pages. fix error in proof

We prove that class NP ≠ class co-NP.

**Category:** Combinatorics and Graph Theory

[149] **viXra:1809.0504 [pdf]**
*submitted on 2018-09-24 14:01:48*

**Authors:** Edigles Guedes

**Comments:** 11 Pages.

I deduce some relations for the q-Pochhammer symbol, the q-binomial coefficient, the q-bracket, the q-factorial and the q-Gamma function.

**Category:** Combinatorics and Graph Theory

[148] **viXra:1809.0188 [pdf]**
*submitted on 2018-09-09 10:39:33*

**Authors:** John Archie Gillis

**Comments:** 53 Pages.

The Clique Problem
This paper provides a Polynomial Time and Non-Heuristic Solution to the Clique problem.
Methods are given to find:
1.Maximum clique (a clique with the largest possible number of vertices),
2.Listing all maximal cliques (cliques that cannot be enlarged), and
3.Solving the decision problem of testing whether a graph contains a clique larger than a given size.
4.Finding cliques of a selected size, particularly largest cliques.

**Category:** Combinatorics and Graph Theory

[147] **viXra:1809.0050 [pdf]**
*submitted on 2018-09-02 10:40:32*

**Authors:** Marco Ripà

**Comments:** 9 Pages.

This paper offers the first proof that the minimal solution of the n x n dots puzzle, for any n≥3, counts 2n-2 lines. Furthermore, a general criterion to solve any n x n grid is given.

**Category:** Combinatorics and Graph Theory

[146] **viXra:1808.0167 [pdf]**
*submitted on 2018-08-13 12:38:39*

**Authors:** Sam Micheal

**Comments:** 1 Page.

game invented by a super-math-genius XOR God inspired it; you decide

**Category:** Combinatorics and Graph Theory

[145] **viXra:1807.0486 [pdf]**
*submitted on 2018-07-28 07:02:51*

**Authors:** Yair Lavi

**Comments:** 5 Pages.

We formulate conjectures regarding the maximum value and maximizing matrices
of the permanent and of diagonal products on the set of stochastic matrices with
bounded rank. We formulate equivalent conjectures on upper bounds for these func-
tions for nonnegative matrices based on their rank, row sums and column sums.
In particular we conjecture that the permanent of a singular nonnegative matrix is
bounded by 1/2 times the minimum of the product of its row sums and the product of
its column sums, and that the product of the elements of any diagonal of a singular
nonnegative matrix is bounded by 1/4 times the minimum of the product of its row
sums and the product of its column sums.

**Category:** Combinatorics and Graph Theory

[144] **viXra:1807.0384 [pdf]**
*submitted on 2018-07-23 09:53:39*

**Authors:** Marco Ripà, Valerio Bencini

**Comments:** 15 Pages.

In this paper we describe two new patterns, in order to improve the upper bound for the Ripà’s n X n X n points problem, saving a few lines for many values of n. The new upper bound, for any n≥6, becomes h_u(n)=int((3/2*n^2)+int(n/4)-int((n-1)/4)+int((n+1)/4)-int((n+2)/4)+n-2, where int(x)≔floor(x).

**Category:** Combinatorics and Graph Theory

[143] **viXra:1807.0114 [pdf]**
*submitted on 2018-07-04 09:45:17*

**Authors:** Said Broumi, Assia Bakali, Mohamed Talea, Florentin Smarandache, V. Venkateswara Rao

**Comments:** 20 Pages. Neutrosophic Operational Research Volume III

The neutrosophic set theory, proposed by smarandache, can be used as a general mathematical tool for dealing with indeterminate and inconsistent information. By applying the concept of neutrosophic sets on graph theory, several studies of neutrosophic models have been presented in the literature. In this paper, the concept of complex neutrosophic graph of type 1 is extended to interval complex neutrosophic graph of type 1(ICNG1). We have proposed a representation of ICNG1 by adjacency matrix and studied some
properties related to this new structure. The concept of ICNG1 generalized the concept of generalized fuzzy graphs of type 1 (GFG1), generalized single valued neutrosophic graphs of type 1 (GSVNG1) generalized interval valued neutrosophic graphs of type 1 (GIVNG1) and complex neutrosophic graph type 1(CNG1).

**Category:** Combinatorics and Graph Theory

[142] **viXra:1806.0336 [pdf]**
*submitted on 2018-06-22 07:08:11*

**Authors:** Thinh Nguyen

**Comments:** 27 Pages.

A polynomial algorithm is “faster” than an exponential algorithm. As n grows an (exponential) always grows faster than nk (polynomial), i.e. for any values of a and k, after n> certain integer n0, it is true that an > nk. Even 2^n grows faster than n1000 at some large value of n. The former functions are exponential and the later functions are polynomial. It seems that for some problems we just may not have any polynomial algorithm at all (as in the information theoretic bound)! The theory of NPcompleteness is about this issue, and in general the computational complexity theory addresses it.

**Category:** Combinatorics and Graph Theory

[141] **viXra:1806.0238 [pdf]**
*submitted on 2018-06-18 00:11:02*

**Authors:** Thinh D. Nguyen

**Comments:** 2 Pages.

The study of perebor dates back to the Soviet-era mathematics, especially in the 1980s [1]. Post-Soviet mathematicians have been working on many problems in combinatorial optimization. One of them is Maximum Edge Biclique Problem (MBP). In [2], the author proves that MBP is NP-complete. In this note, we give a polynomial time algorithm for MBP by using linear programming (LP). Thus, some flaw needs to be found in Peeter's work. We leave this to the community.

**Category:** Combinatorics and Graph Theory

[140] **viXra:1806.0179 [pdf]**
*submitted on 2018-06-14 03:30:42*

**Authors:** Thinh D. Nguyen

**Comments:** 2 Pages.

The study of perebor dates back to the Soviet-era mathematics, especially in the 1980s [1]. Post-Soviet mathematicians has been working on many problems in combinatorial optimization. One of them is Exact Weight Perfect Matching of Bipartite Graph (EWPM).This particular problem has been thoroughly considered by [2], [3], [4]. In this note, we give a simpler proof about the solvability of EWPM.

**Category:** Combinatorics and Graph Theory

[139] **viXra:1806.0065 [pdf]**
*submitted on 2018-06-05 08:32:38*

**Authors:** Zhuwenhao

**Comments:** 3 Pages.

This proves that any even number larger than 2 can be written as the sum of two prime Numbers, also known as the "goldbach conjecture" or "goldbach conjecture about the even" is in the test for any greater than or equal to 6 even conform to guess the number of prime Numbers, accidentally discovered the prime Numbers of "additionality" and further expansion of verification.This article does not focus on the functional expressions of prime Numbers themselves, but takes a different approach to prove that all even Numbers can be composed of two prime Numbers

**Category:** Combinatorics and Graph Theory

[138] **viXra:1806.0049 [pdf]**
*submitted on 2018-06-06 02:48:22*

**Authors:** Thinh D. Nguyen

**Comments:** 10 Pages.

In this paper, we present several models for Capacitated Vehicle Routing Problem (CVRP) using Choco solver. A concise introduction to the constraint programming methods is included. Then, we construct two models for CVRP. Experimental results for each model are given in details.

**Category:** Combinatorics and Graph Theory

[137] **viXra:1806.0045 [pdf]**
*submitted on 2018-06-04 06:05:03*

**Authors:** Thinh Duc Nguyen

**Comments:** 1 Page.

In this note, we show that graph isomorphism is reducible to circuit isomorphism, in polynomial time.

**Category:** Combinatorics and Graph Theory

[136] **viXra:1805.0377 [pdf]**
*submitted on 2018-05-22 20:28:44*

**Authors:** Prashanth R. Rao

**Comments:** 2 Pages.

Abstract: In this paper we show a method to encode graphs with a numerical value that follows unique labeling of each vertex or node and unique labeling of each edge of a graph with unique prime numbers. Each edge is defined as the connectivity between two vertices, therefore two vertices or nodes connected by an edge may be represented by the “ edge-nodes value ” derived by raising the prime number representing the edge to the product of the primes representing the two nodes that are connected by that edge. Multiplying all the “edge-nodes values” of a single graph will represent a unique number albeit very large in majority of cases. Given this unique number called the “Edge-nodes values product”, it is possible to derive the structure of the given graph. This encoding may allow new approaches to graph isomorphism, cryptography, quantum computing, data security, artificial intelligence, etc.

**Category:** Combinatorics and Graph Theory

[135] **viXra:1805.0205 [pdf]**
*submitted on 2018-05-10 10:33:08*

**Authors:** Richard J. Mathar

**Comments:** 70 Pages.

The non-cyclic graphs known as trees may be labeled by assigning
positive integer numbers (weights) to their vertices or to their edges.
We count the trees
up to 10 vertices that have prescribed sums of weights, or, from
the number-theoretic point of view, we count the compositions
of positive integers that are constrained by
the symmetries of trees.

**Category:** Combinatorics and Graph Theory

[134] **viXra:1804.0172 [pdf]**
*submitted on 2018-04-12 14:37:14*

**Authors:** Said Broumi, Assia Bakali, Mohamed Talea, Florentin Smarandache, Kifayat Ullah

**Comments:** 6 Pages. Broumi, Said and Talea, Mohamed and Bakali, Assia and Smarandache, Florentin and Ullah, Kifayat, Bipolar Neutrosophic Minimum Spanning Tree (February 27, 2018). Smart Application and Data Analysis for Smart Cities (SADASC'18). Available at SSRN: https://s

The aim of this article is to introduce a matrix algorithm for finding minimum spanning tree (MST) in the environment of undirected bipolar neutrosophic connected graphs (UBNCG). Some weights are assigned to each edge in the form of bipolar neutrosophic number (BNN). The new algorithm is described by a flow chart and a numerical example by considering some hypothetical graph. By a comparison, the advantage of proposed matrix algorithm over some existing algorithms are also discussed.

**Category:** Combinatorics and Graph Theory

[133] **viXra:1804.0165 [pdf]**
*submitted on 2018-04-12 10:43:40*

**Authors:** Zbigniew Osiak

**Comments:** 156 Pages. In Polish

Rozpatrywane w tej pracy rodzaje grafów można utworzyć na podstawie równań stechiometrycznych. Główną zaletą metod grafowych jest możliwość uzyskania informacji o stabilności stanów stacjonarnych bez wypisywania w jawnej postaci jakichkolwiek równań.
###
The types of graphs considered in this work can be created on the basis of stoichiometric equations. The main advantage of the graph theory methods is the possibility of obtaining information on the stability of steady states without writing out any equations in an explicit form.

**Category:** Combinatorics and Graph Theory

[132] **viXra:1803.0163 [pdf]**
*submitted on 2018-03-12 03:56:18*

**Authors:** Juan Moreno Borrallo

**Comments:** 4 Pages.

In this paper it is discussed the following problem: "A mathematician
wants to divide is garden into consecutive prime parts (first in two parts,
after in three parts, and so on), only making straight paths, in a simple
way (without retracing his own steps), and without going out of his plot
of land. In how many parts can the mathematician divide his garden?"

**Category:** Combinatorics and Graph Theory

[131] **viXra:1802.0393 [pdf]**
*submitted on 2018-02-27 00:34:27*

**Authors:** Sudhir Jog, Shrinath Patil

**Comments:** 9 Pages.

Ramane,Yalnaik recently defined some molecular structural descriptors on the lines of Weiner index, Zagreb Index, etc known as status indices, harmonic status indices, status coindices and harmonic status coindices. Here we consider some graphs of fixed diameter and compute the all these parameters.

**Category:** Combinatorics and Graph Theory

[130] **viXra:1802.0278 [pdf]**
*submitted on 2018-02-20 06:18:10*

**Authors:** Ero Sukna

**Comments:** 7 Pages.

Timestamp : 20/Feb/2018

**Category:** Combinatorics and Graph Theory

[129] **viXra:1801.0117 [pdf]**
*submitted on 2018-01-09 13:10:24*

**Authors:** Arturo Tozzi

**Comments:** 10 Pages.

The assessment of hidden causal relationships, e.g., adverse drug reactions in pharmacovigilance, is currently based on rather qualitative parameters. In order to find more quantifiable parameters able to establish the validity of the alleged correlations between drug intake and onset of symptoms, we introduce the Borsuk-Ulam Theorem (BUT), which states that a single point on a circumference projects to two points on a sphere. The BUT stands for a general principle that describes issues from neuroscience, theoretical physics, nanomaterials, computational topology, chaotic systems, group theory, cosmology. Here we introduce a novel BUT variant, termed operational-BUT, that evaluates causal relationships. Further, we demonstrate that the BUT is correlated with graph theory and in particular with the so-called Kneser graphs: this means that the combinatory features of observables, such as the bodily responses to drug intake, can be described in terms of dynamical mappings and paths taking place on well-established abstract structures. Therefore, physical and biological dynamical systems (including alleged causes and their unknown effects) make predictable moves into peculiar phase spaces, giving rise to
constrained trajectories that can be quantified.

**Category:** Combinatorics and Graph Theory

[128] **viXra:1801.0115 [pdf]**
*submitted on 2018-01-09 18:46:32*

**Authors:** Vixraisshit Usearxiv

**Comments:** 85 Pages. why are you still reading this? go to arXiv!!!

viXra is shitt
go use arXiv
if i can publish bs liek dis den why tf shood you trust anything on dis cite
P.S. I'm not illiterate, it's to prove my point.

**Category:** Combinatorics and Graph Theory

[127] **viXra:1711.0432 [pdf]**
*submitted on 2017-11-27 01:02:26*

**Authors:** Arsen A. Movsesyan

**Comments:** 23 Pages.

In the article, the Gauss’s problem on the number of integer points for a circle and a ball in the framework of an integer lattice is reformulated in an equivalent way and reduces to solving two combinatorial tasks for a circular and spherical "layer" in the framework of Quantum Discrete Space. These tasks are solved using trigonometric functions defined on a set of integers whose range of values is also integers, and other new mathematical tools. It comes not about evaluative solutions, but about exact solutions, which, if necessary, can be transferred to a circle and a ball. In doing so not only specific formulas for determine the exact number of solutions are presented, but also the formulas for enumerating the corresponding pairs and triples of integers. The importance of obtained solutions lies in the fact that they determine the analytical likenesses of not only the circumference and the sphere in the Quantum Discrete Space, but also point to the possibility of constructing of the likenesses of ellipse, cone, hyperboloid and other figures.

**Category:** Combinatorics and Graph Theory

[126] **viXra:1711.0252 [pdf]**
*submitted on 2017-11-09 01:59:34*

**Authors:** Murugesan, Anitha

**Comments:** 9 Pages. The last theorem is the highlight of the paper

In the graph theory, two graphs are said to be isomorphic if there is a one-one, onto mapping defined between their set of vertices so as to preserving the adjacency between vertices. An isomorphism defined on a vertex set of a graph to itself is called automorphism of the given graph. Two vertices in a graph are said to be similar if there is an automorphism defined on its vertex set mapping one vertex to the other. In this paper, it has been discussed that every such automorphism defines an equivalence relation on the set of vertices and the number of equivalence classes is same as the number of rotations that the automorphism makes on the vertex set. The set of all automorphisms of a graph is a permutation group under the composition of permutations. This group is called automorphism group of the graph. A graph is said to be vertex transitive if its automorphism group acts transitively on its vertex set. The path degree sequence of a vertex in a graph is the list of lengths of paths having this particular vertex as initial vertex. The ordered set of all such sequences is called path degree sequence of the graph. It is conjectured that two graphs are isomorphic iff they have same path degree sequence. In this paper, it has been discussed that this conjecture holds good when both the graphs are vertex transitive. The notion of functional graph has been introduced in this paper. The functional graph of any two isomorphic graphs is a graph in which the vertex set is the union of vertex sets of isomorphic graphs and two vertices are connected by an edge iff they are connected in any one of the graph when they belong to the same graph or one vertex is the image of the other under the given isomorphism when they are in different set of vertices. It has been proved that the functional graphs obtained from two isomorphic complete bipartite graphs are vertex transitive.
Keywords : graph automorphism ; functional graph ; vertex transitive graph ; path degree sequence.

**Category:** Combinatorics and Graph Theory

[125] **viXra:1710.0248 [pdf]**
*submitted on 2017-10-22 16:34:08*

**Authors:** Paris Samuel Miles-Brenden

**Comments:** 2 Pages. None.

Note on Combinatorics.

**Category:** Combinatorics and Graph Theory

[124] **viXra:1709.0421 [pdf]**
*submitted on 2017-09-29 00:08:36*

**Authors:** Choe Ryujin

**Comments:** 6 Pages.

Simple solution of traveling salesman problem

**Category:** Combinatorics and Graph Theory

[123] **viXra:1709.0064 [pdf]**
*submitted on 2017-09-06 05:06:03*

**Authors:** Said Broumi, Assia Bakali, Mohamed Talea, Ali Hassan, Florentin Smarandache

**Comments:** 7 Pages. 2017 IEEE International Conference on INnovations in Intelligent SysTems and Applications (INISTA), Gdynia Maritime University, Gdynia, Poland, 3-5 July 2017.

In this paper, motivated by the notion of
generalized single valued neutrosophic graphs of first type, we
defined a new neutrosophic graphs named generalized interval
valued neutrosophic graphs of first type (GIVNG1) and
presented a matrix representation for it and studied few
properties of this new concept. The concept of GIVNG1 is an
extension of generalized fuzzy graphs (GFG1) and generalized
single valued neutrosophic of first type (GSVNG1).

**Category:** Combinatorics and Graph Theory

[122] **viXra:1709.0063 [pdf]**
*submitted on 2017-09-06 05:08:06*

**Authors:** Said Broumi, Assia Bakali, Mohamed Talea, Florentin Smarandache

**Comments:** 6 Pages. 2017 IEEE International Conference on INnovations in Intelligent SysTems and Applications (INISTA), Gdynia Maritime University, Gdynia, Poland, 3-5 July 2017.

In this article, we present an algorithm method
for finding the shortest path length between a paired nodes on a
network where the edge weights are characterized by single
valued triangular neutrosophic numbers. The proposed
algorithm gives the shortest path length from source node to
destination node based on a ranking method. Finally a numerical
example is presented to illustrate the efficiency of the proposed
approach

**Category:** Combinatorics and Graph Theory

[121] **viXra:1709.0062 [pdf]**
*submitted on 2017-09-06 05:14:18*

**Authors:** Said Broumi, Assia Bakali, Mohamed Talea, Florentin Smarandache

**Comments:** 6 Pages. 2017 IEEE International Conference on INnovations in Intelligent SysTems and Applications (INISTA), Gdynia Maritime University, Gdynia, Poland, 3-5 July 2017

In this paper, we introduced a new neutrosophic graphs called complex neutrosophic graphs of type1 (CNG1) and presented a matrix representation for it and studied some properties of this new concept. The concept of CNG1 is an extension of generalized fuzzy graphs of type 1 (GFG1) and generalized single valued neutrosophic graphs of type 1 (GSVNG1).

**Category:** Combinatorics and Graph Theory

[120] **viXra:1708.0404 [pdf]**
*submitted on 2017-08-28 06:04:10*

**Authors:** Richard J. Mathar

**Comments:** Pages 47-120 are adjacency matrices for connected simple graphs up to 7 nodes.

We create the unlabeled or vertex-labeled graphs with up to 10
edges and up to 10 vertices and classify them by a set of standard properties:
directed or not, vertex-labeled or not, connectivity, presence of isolated vertices, presence of multiedges and presence of loops. We present tables of how
many graphs exist in these categories.

**Category:** Combinatorics and Graph Theory

[119] **viXra:1708.0163 [pdf]**
*submitted on 2017-08-15 02:58:11*

**Authors:** A. A. Frempong

**Comments:** 12 Pages. Copyright © by A. A. Frempong

The traveling salesman can determine by hand, with zero or negligible error, the shortest route from home base city to visit once, each of three cities, 10 cities, 20 cities, 100 cities, or 1000 cities, and return to the home base city. The general approach to solving the different types of NP problems is the same, except that sometimes, specific techniques may differ from each other according to the process involved in the problem. In the salesman problem, the first step is to arrange the data in the problem in increasing order, since one's interest is in the shortest distances. The main principle here is that the shortest route is the sum of the shortest distances such that the salesman visits each city once and returns to the starting city. The approach in this paper is different from the author's previous approach (viXra:1505.0167) in which the needed distances not among the least ten distances were added to the least ten distances before route construction began. In this paper, one starts with only the least ten distances and only if a needed distance is not among the set of the least ten distances, would one consider distances greater than those in the set of the ten least distances.
The shortest route to visit nine cities and return to the starting city was found in this paper. It was also found out that even though the length of the shortest route is unique, the sequence of the cities involved is not unique. The approach used in this paper can be applied in workforce project management and hiring, as well as in a country's workforce needs and immigration quota determination. Since an approach that solves one of these problems can also solve other NP problems, and the traveling salesman problem has been solved, all NP problems can be solved, provided that one has an open mind and continues to think. If all NP problems can be solved, then all NP problems are P problems and therefore, P is equal to NP. The CMI Millennium Prize requirements have been satisfied

**Category:** Combinatorics and Graph Theory

[118] **viXra:1707.0298 [pdf]**
*submitted on 2017-07-23 11:06:26*

**Authors:** Marco Ripà

**Comments:** This is a revised version of the paper published in 2016 on Notes on Number Theory and Discrete Mathematics (ISSN 1310-5132), Volume 22, Number 2 (Pages 36—43).

We provide an optimal strategy to solve the n X n X n points problem inside the box, considering only 90° turns, and at the same time a pattern able to drastically lower down the known upper bound. We use a very simple spiral frame, especially if compared to the previous plane by plane approach, that significantly reduces the number of straight lines connected at their end-points necessary to join all the n^3 dots. In the end, we combine the square spiral frame with the rectangular spiral pattern in the most profitable way, in order to minimize the difference between the upper and the lower bound, proving that it is ≤ 0.5 ∙ n ∙ (n + 3), for any n > 1.

**Category:** Combinatorics and Graph Theory

[117] **viXra:1701.0341 [pdf]**
*submitted on 2017-01-10 02:57:03*

**Authors:** Alexey V. Komkov

**Comments:** 2 Pages.

This work contains certificates numbers Van der Waerden, was found using SAT Solver. These certificates establish the best currently known lower bounds of the numbers Van der Waerden W( 7, 3 ), W( 8, 3 ), W( 9, 3 ), W( 10, 3 ), W( 11, 3 ).

**Category:** Combinatorics and Graph Theory

[116] **viXra:1612.0248 [pdf]**
*submitted on 2016-12-14 10:54:53*

**Authors:** Vakkas Uluçay, Mehmet Şahin, Said Broumi, Assia Bakali, Mohamed Talea, Florentin Smarandache

**Comments:** 30 Pages. this article is submited to an elsevier journal

In this paper, we first define the concept of neutrosophic soft expert graph. We have established a link between graphs and neutrosophic soft expert sets. Basic operations of neutrosophic soft expert graphs such as union, intersection and complement are defined here. The concept of neutrosophic soft expert soft graph is also discussed in this paper. The new concept is called neutrosophic soft expert graph-based multi-criteria decision making method (NSEGMCDM for short). Finally, an illustrative example is given and a comparison analysis is conducted between the proposed approach and other existing methods, to verify the feasibility and effectiveness of the developed approach.

**Category:** Combinatorics and Graph Theory

[115] **viXra:1611.0385 [pdf]**
*submitted on 2016-11-28 14:44:16*

**Authors:** Alexey V. Komkov

**Comments:** Pages.

Genetic algorithm is a good tool for finding the global minimum in many discrete problems. In particular, it has proven itself in problems where there is no any apriori information about the possibilities of narrowing the search, or the specifics of the problem do not involve such. This work describes the procedure of using a genetic algorithm as applied to the search of van der Waerden numbers. Some new lower bounds of van der Waerden numbers were found using this procedure.

**Category:** Combinatorics and Graph Theory

[114] **viXra:1611.0327 [pdf]**
*submitted on 2016-11-23 16:16:31*

**Authors:** Shimaa Fathi, Hewayda Elghawalby, A.a. Salama

**Comments:** 8 Pages.

This paper is devoted for presenting new neutrosophic similarity measures between
neutrosophic graphs. We proposetwo ways to determine the neutrosophic distance between
neutrosophic vertex graphs. The two neutrosophic distances are based on the Haussdorff distance,and a robust modified variant of the Haussdorff distance, moreover we show that they both satisfy the metric distance measure axioms. Furthermore, a similarity measure between neutrosophic edge graphs that is based on a probabilistic variant of Haussdorff distance is introduced. The aim is to use those measures for the purpose of matching neutrosophic graphs whose structure can be described in the neutrosophic domain.

**Category:** Combinatorics and Graph Theory

[113] **viXra:1611.0324 [pdf]**
*submitted on 2016-11-24 02:58:02*

**Authors:** Eman.m.el-Nakeeb, H. Elghawalby, A.a.salama, S.a.el-Hafeez

**Comments:** 17 Pages.

The aim of this paper is to introduce a new approach to Mathematical Morphology based on
neutrosophic set theory. Basic definitions for neutrosophic morphological operations are extracted and a study of its algebraic properties is presented. In our work we demonstrate that neutrosophic morphological operations inherit properties and restrictions of Fuzzy Mathematical Morphology

**Category:** Combinatorics and Graph Theory

[112] **viXra:1610.0050 [pdf]**
*submitted on 2016-10-04 21:54:55*

**Authors:** Renny P Varghese, K Rejikumar

**Comments:** 8 Pages.

Let G1 and G2 be two graph with vertex sets V (G1); V (G2) and
edge sets E(G1);E(G2) respectively. The subdivision graph S(G) of a graph
G is the graph obtained by inserting a new vertex into every edges of G. The
SGvertexjoin of G1 and G2 is denoted by G1}G2 and is the graph obtained
from S(G1) [ G1 and G2 by joining every vertex of V (G1) to every vertex
of V (G2). In this paper we determine the adjacency spectra ( respectively
Laplacian spectra and signless Laplacian spectra) of G1}G2 for a regular graph
G1 and an arbitrary graph G2

**Category:** Combinatorics and Graph Theory

[111] **viXra:1610.0049 [pdf]**
*submitted on 2016-10-04 22:15:42*

**Authors:** K. Reji Kumar, Renny P. Varghese

**Comments:** 10 Pages.

The Duplication graph DG of a graph G, is obtained by inserting
new vertices corresponding to each vertex of G and making the vertex adja-
cent to the neighbourhood of the corresponding vertex of G and deleting the
edges of G. Let G1 and G2 be two graph with vertex sets V (G1) and V (G2)
respectively. The DG - vertex join of G1 and G2 is denoted by G1 t G2 and
it is the graph obtained from DG1 and G2 by joining every vertex of V (G1)
to every vertex of V (G2). The DG - add vertex join of G1 and G2 is denoted
by G1 ./ G2 and is the graph obtained from DG1 and G2 by joining every
additional vertex of DG1 to every vertex of V (G2). In this paper we determine
the A - spectra and L - spectra of the two new joins of graphs for a regular
graph G1 and an arbitrary graph G2 . As an application we give the number
of spanning tree, the Kirchhoff index and Laplace energy like invariant of the
new join. Also we obtain some infinite family of new class of integral graphs

**Category:** Combinatorics and Graph Theory

[110] **viXra:1610.0043 [pdf]**
*submitted on 2016-10-04 11:30:07*

**Authors:** K Reji Kumar, Renny P Varghese

**Comments:** 8 Pages.

We present a spectral theory of uniform, regular and linear hyper-
graph. The main result are the nature of the eigen values of (k; r) - regular
linear hypergraph and the relation between its dual and line graph. We also
discuss some properties of Laplacian spectrum of a (k; r) - regular hypergraphs.

**Category:** Combinatorics and Graph Theory

[109] **viXra:1609.0113 [pdf]**
*submitted on 2016-09-09 05:47:27*

**Authors:** Damodar Rajbhandari

**Comments:** 9 Pages.

In this paper, we'll be discussing about the "Seven Bridges of Königsberg Problem". This paper will help you to understand, "How Euler solved this problem?". And, It will give some intuition of "Topology" and "Graph Theory".

**Category:** Combinatorics and Graph Theory

[108] **viXra:1608.0380 [pdf]**
*submitted on 2016-08-28 11:00:45*

**Authors:** Richard J. Mathar

**Comments:** 10 pages are Java source code distributed under the LGPL 3.

This is a numerical study of the combinatorial problem of packing hexagons of some equal size
into a larger hexagon. The problem is well defined if all hexagon edges have integer length
and if their centers and vertices share the common lattice points of a triangular grid with unit distances.

**Category:** Combinatorics and Graph Theory

[107] **viXra:1606.0074 [pdf]**
*submitted on 2016-06-07 19:53:07*

**Authors:** Lucas Allen

**Comments:** English, 9 pages, matrix equations and examples

This article presents an algorithm for solving the graph isomorphism problem. Under certain circumstances the algorithm is definitely polynomial time, and it could possibly always be polynomial time, but that hasn't been verified. The algorithm also hasn't been tested on graphs with more than three nodes, nor has it been reviewed by anyone so far.

**Category:** Combinatorics and Graph Theory

[106] **viXra:1605.0213 [pdf]**
*submitted on 2016-05-20 20:24:35*

**Authors:** Michail Zak

**Comments:** 15 Pages.

The challenge of this paper is to relate artificial intuition-based intelligence, represented by self-supervised systems, to solutions of NP-complete problems. By self-supervised systems we understand systems that are capable to move from disorder to order without external effort, i.e. in violation of the second law of thermodynamics. It has been demonstrated, [1], that such systems exist in the mathematical world: they are presented by ODE coupled with their Liouville equation, but they belong neither to Newtonian nor to quantum physics since they are capable to violate the second law of thermodynamics. That suggests that machines could not simulate intuition-based intelligence if they are composed only of physical parts, but without digital components. Nevertheless it was found such quantum-classical hybrids, [1], that simulates some of self-supervised systems. The main achievement of this work is a demonstration that self-supervised systems can solve NP-complete problems in polynomial time by replacing an enumeration of exponentially large number of possible choices with a short cut provided by a non-Newtonian and non-quantum nature of self-supervised systems.

**Category:** Combinatorics and Graph Theory

[81] **viXra:1911.0203 [pdf]**
*replaced on 2019-12-22 02:44:16*

**Authors:** A. A. Frempong

**Comments:** 38 Pages. Copyright © by A. A. Frempong

The solution of the traveling salesman problem (TSP) in this paper makes this problem no longer an NP-hard problem, but rather, a P problem. The TSP was solved in polynomial time and its solution was also correctly checked in polynomial time. Also solved were an NP-Complete TSP, and six other NP-Complete problems. The TSP solution killed two (three) birds with one stone, because its solution made the NP-hard problems and NP-Complete problems become P problems. The shortest route as well as the longest route for the salesman to visit each of nine cities once and return to the base city was determined. In finding the shortest route, the first step was to arrange the data of the problem in increasing order, since one's interest is in the shortest distances; but in finding the longest route, the first step was to arrange the data of the problem in decreasing order, since one's interest is in the longest distances. For the shortest route, the main principle is that the shortest route is the sum of the shortest distances such that the salesman visits each city once and returns to the starting city; but for the longest route, the main principle is that the longest route is the sum of the longest distances such that the salesman visits each city once and returns to the starting city. Since ten cities are involved, ten distances would be needed for the salesman to visit each of nine cities once and return to the starting city. One started the construction of the shortest route using only the shortest ten distances; and if a needed distance was not among the set of the shortest ten distances, one would consider distances longer than those in the set of the shortest ten distances. For the longest route, the construction began using only the longest ten distances; and if a needed distance was not among the set of the longest ten distances, one would consider distances shorter than those in the set of the longest ten distances. It was found out that even though, the length of the shortest or the longest route is unique, the sequence of the cities involved is not unique. The approach used in this paper can be applied in work-force project management and hiring, as well as in a country's work-force needs and immigration quota determination. Since approaches that solve the TSP and NP-Complete problems can also solve other NP problems, and the TSP and NP-Complete problems have been solved, all NP problems can be solved. If all NP problems can be solved, then all NP problems are P problems, and therefore, P is equal to NP. The CMI Millennium Prize requirements have been satisfied.

**Category:** Combinatorics and Graph Theory

[80] **viXra:1911.0203 [pdf]**
*replaced on 2019-11-12 02:43:47*

**Authors:** A. A. Frempong

**Comments:** 38 Pages. Copyright © by A. A. Frempong

The solution of the traveling salesman problem (TSP) in this paper makes this problem no longer an NP-hard problem, but rather, a P problem. The TSP was solved in polynomial time and its solution was also correctly checked in polynomial time. Also solved were an NP-Complete TSP, and six other NP-Complete problems. The TSP solution killed two (three) birds with one stone, because its solution made the NP-hard problems and NP-Complete problems become P problems. The shortest route as well as the longest route for the salesman to visit each of nine cities once and return to the base city was determined. In finding the shortest route, the first step was to arrange the data of the problem in increasing order, since one's interest is in the shortest distances; but in finding the longest route, the first step was to arrange the data of the problem in decreasing order, since one's interest is in the longest distances. For the shortest route, the main principle is that the shortest route is the sum of the shortest distances such that the salesman visits each city once and returns to the starting city; but for the longest route, the main principle is that the longest route is the sum of the longest distances such that the salesman visits each city once and returns to the starting city. Since ten cities are involved, ten distances would be needed for the salesman to visit each of nine cities once and return to the starting city. One started the construction of the shortest route using only the shortest ten distances; and if a needed distance was not among the set of the shortest ten distances, one would consider distances longer than those in the set of the shortest ten distances. For the longest route, the construction began using only the longest ten distances; and if a needed distance was not among the set of the longest ten distances, one would consider distances shorter than those in the set of the longest ten distances. It was found out that even though, the length of the shortest or the longest route is unique, the sequence of the cities involved is not unique. The approach used in this paper can be applied in work-force project management and hiring, as well as in a country's work-force needs and immigration quota determination. Since approaches that solve the TSP and NP-Complete problems can also solve other NP problems, and the TSP and NP-Complete problems have been solved, all NP problems can be solved. If all NP problems can be solved, then all NP problems are P problems, and therefore, P is equal to NP. The CMI Millennium Prize requirements have been satisfied

**Category:** Combinatorics and Graph Theory

[79] **viXra:1910.0283 [pdf]**
*replaced on 2019-11-11 20:44:29*

**Authors:** Yasushi Ieno

**Comments:** 17 Pages.

Given two sets, one consisting of variables representing distinct positive n numbers, the other set `a kind of power set' of this n-element set. I got interested in the fact that for the latter set, depending on the values of two elements, it can occur that not every pair of elements is `comparable', that is to say, it is not always uniquely determined which of two elements is smaller. By proving theorems in order to go ahead with our research, we show a table which describes for how many `comparable' cases exist, for several n's.

**Category:** Combinatorics and Graph Theory

[78] **viXra:1910.0283 [pdf]**
*replaced on 2019-10-27 07:38:17*

**Authors:** Yasushi Ieno

**Comments:** 16 Pages.

Given two sets, one consisting of variables representing distinct positive n numbers, the other set `a kind of power set' of this n-element set. I got interested in the fact that for the latter set, depending on the values of two elements, it can occur that not every pair of elements is `comparable', that is to say, it is not always uniquely determined which of two elements is smaller. By proving theorems in order to go ahead with our research, we show a table which describes for how many 'comparable' cases exist, for several n's.

**Category:** Combinatorics and Graph Theory

[77] **viXra:1910.0283 [pdf]**
*replaced on 2019-10-17 02:51:02*

**Authors:** Yasushi Ieno

**Comments:** 16 Pages.

Given two sets, one consisting of variables representing distinct positive n numbers, the other set `a kind of power set' of this n-element set. I got interested in the fact that for the latter set, depending on the values of two elements, it can occur that not every pair of elements is `comparable', that is to say, it is not always uniquely determined which of two elements is smaller. By proving theorems in order to go ahead with our research, we show a table which describes for how many `comparable' cases exist, for several n's.

**Category:** Combinatorics and Graph Theory

[76] **viXra:1910.0230 [pdf]**
*replaced on 2019-11-03 22:12:39*

**Authors:** Yasushi Ieno

**Comments:** 31 Pages.

The 6th problem of the 50th International Mathematical Olympiad (IMO), held in Germany, 2009, is called 'the grasshopper problem'. To this problem Kos developed theory from unique viewpoints by reference of Noga Alon’s combinatorial Nullstellensatz.
We have tried to solve this problem by an original method inspired by a polynomial function that Kos defined, then examined for n=3, 4 and 5. For almost cases the claim of this problem follows, but there remains imperfection due to 'singularity'.

**Category:** Combinatorics and Graph Theory

[75] **viXra:1910.0230 [pdf]**
*replaced on 2019-10-30 21:47:15*

**Authors:** Yasushi Ieno

**Comments:** 30 Pages.

The 6th problem of the 50th International Mathematical Olympiad (IMO), held in Germany, 2009, is called 'the grasshopper problem'. To this problem Kos developed theory from unique viewpoints by reference of Noga Alon’s combinatorial Nullstellensatz.
We have tried to solve this problem by an original method inspired by a polynomial function that Kos defined, then examined for n=3, 4 and 5. For almost cases the claim of this problem follows, but there remains imperfection due to 'singularity'.

**Category:** Combinatorics and Graph Theory

[74] **viXra:1910.0230 [pdf]**
*replaced on 2019-10-28 02:28:28*

**Authors:** Yasushi Ieno

**Comments:** 28 Pages.

The 6th problem of the 50th International Mathematical Olympiad (IMO), held in Germany, 2009, is called 'the grasshopper problem'. To this problem Kos developed theory from unique viewpoints by reference of Noga Alon’s combinatorial Nullstellensatz.
We have tried to solve this problem by an original method inspired by a polynomial function that Kos defined, then examined for n=3, 4 and 5. For almost cases the claim of this problem follows, but there remains imperfection due to 'singularity'.

**Category:** Combinatorics and Graph Theory

[73] **viXra:1910.0230 [pdf]**
*replaced on 2019-10-26 00:23:19*

**Authors:** Yasushi Ieno

**Comments:** 28 Pages.

The 6th problem of the 50th International Mathematical Olympiad (IMO), held in Germany, 2009, is called 'the grasshopper problem'. To this problem Kos developed theory from unique viewpoints by reference of Noga Alon’s combinatorial Nullstellensatz.
We have tried to solve this problem by an original method inspired by a polynomial function that Kos defined, then researched for n=3, 4 and 5. For almost cases the claim of this problem follows, but there remains imperfection due to 'singularity'.

**Category:** Combinatorics and Graph Theory

[72] **viXra:1910.0230 [pdf]**
*replaced on 2019-10-23 06:48:32*

**Authors:** Yasushi Ieno

**Comments:** 27 Pages.

The 6th problem of the 50th International Mathematical Olympiad (IMO), held in Germany, 2009, is called 'the grasshopper problem'. To this problem Kos developed theory from unique viewpoints by reference of Noga Alon’s combinatorial Nullstellensatz.
We have tried to solve this problem by an original method inspired by a polynomial function that Kos defined, then researched for n=3, 4 and 5. For almost cases the claim of this problem follows, but there remains imperfection due to 'singularity'.

**Category:** Combinatorics and Graph Theory

[71] **viXra:1910.0230 [pdf]**
*replaced on 2019-10-21 05:17:30*

**Authors:** Yasushi Ieno

**Comments:** 25 Pages.

The 6th problem of the 50th International Mathematical Olympiad (IMO), held in Germany, 2009, is called 'the grasshopper problem'. To this problem Kos developed theory from unique viewpoints by reference of Noga Alon’s combinatorial Nullstellensatz.
We have tried to solve this problem by an original method inspired by a polynomial function that Kos defined, then researched for n=3, 4 and 5. For almost cases the problem can be solved, but there remains imperfection due to 'singularity'.

**Category:** Combinatorics and Graph Theory

[70] **viXra:1910.0230 [pdf]**
*replaced on 2019-10-18 02:49:45*

**Authors:** Yasushi Ieno

**Comments:** 25 Pages.

The 6th problem of the 50th International Mathematical Olympiad (IMO), held in Germany, 2009, is called 'the grasshopper problem'. To this problem Kos developed theory from unique viewpoints by reference of Noga Alon’s combinatorial Nullstellensatz.
We have tried to solve this problem by an original method inspired by a polynomial function that Kos defined, then researched for n=3, 4 and 5. For almost cases the problem can be solved, but there remains imperfection due to 'singularity'.

**Category:** Combinatorics and Graph Theory

[69] **viXra:1910.0230 [pdf]**
*replaced on 2019-10-15 06:34:22*

**Authors:** Yasushi Ieno

**Comments:** 7 Pages.

The 6th problem of the 50th International Mathematical Olympiad (IMO), held in Germany, 2009, is called 'the grasshopper problem'. To this problem Kos developed theory from unique viewpoints by reference of Noga Alon's combinatorial Nullstellensatz. We have tried to solve this problem by an original method inspired by a polynomial function that Kos defined, then researched for n=3 and n=4. For almost cases the problem can be solved, but there remains imperfection due to 'singularity'.

**Category:** Combinatorics and Graph Theory

[68] **viXra:1908.0480 [pdf]**
*replaced on 2019-09-03 05:14:40*

**Authors:** A. A. Frempong

**Comments:** 11 Pages. Copyright © by A. A. Frempong

By applying differential and integral calculus, this paper covers the principles and procedures for producing the solution of a problem, given the procedure for checking the correctness of the solution of a problem, and vice versa. If one is able to check quickly and completely, the correctness of the solution of a problem, one should also be able to produce the solution of the problem by reversing the order of the steps of the checking process, while using opposite operations in each step. The above principles were applied to four examples from calculus as well as to an example from geometry. Even though in calculus, one normally uses differentiation to check the correctness of an integration result, one will differentiate a function first, and then integrate the derivative to obtain the original function. One will differentiate the trigonometric functions, tan x, cot x, sec x and csc x; followed by integrating each derivative to obtain each original function. The results show that the solution process and the checking process are inverses of each other. In checking the correctness of the solution of a problem, one should produce the complete checking procedure which includes the beginning, the middle, and the end of the problem. Checking only the correctness of the final answer or statement is incomplete checking. To facilitate complete checking, the question should always be posed such that one is compelled to show a complete checking procedure from which the solution procedure can be produced. A general application of P = NP is that, if the correctness of the solution of a problem can be checked quickly and it is difficult to write a solution procedure, then first, one can write a complete checking procedure and reverse the order of the steps of the checking procedure while using opposite operations in each step, to obtain the solution procedure for the problem. Therefore, P is equal to NP. Furthermore, every problem with a complete procedure for checking the correctness of the solution of the problem can be solved in polynomial time.

**Category:** Combinatorics and Graph Theory

[67] **viXra:1908.0480 [pdf]**
*replaced on 2019-08-31 02:58:43*

**Authors:** A. A. Frempong

**Comments:** 10 Pages. Copyright © by A. A. Frempong

By applying differential and integral calculus, this paper covers the principles and procedures for producing the solution of a problem, given the procedure for checking the correctness of the solution of a problem, and vice versa. If one is able to check quickly and completely, the correctness of the solution of a problem, one should also be able to produce the solution of the problem by reversing the order of the steps of the checking process, while using opposite operations in each step. The above principles were applied to four examples from calculus as well as to an example from geometry. Even though in calculus, one normally uses differentiation to check the correctness of an integration result, one will differentiate a function first, and then integrate the derivative to obtain the original function. One will differentiate the trigonometric functions, tan x, cot x, sec x and csc x; followed by integrating each derivative to obtain each original function. The results show that the solution process and the checking process are inverses of each other. In checking the correctness of the solution of a problem, one should produce the complete checking procedure which includes the beginning, the middle, and the end of the problem. Checking only the correctness of the final answer or statement is incomplete checking. To facilitate complete checking, the question should always be posed such that one is compelled to show a complete checking procedure from which the solution procedure can be produced. A general application of P = NP is that, if the correctness of the solution of a problem can be checked quickly and it is difficult to write a solution procedure, then first, write a complete checking procedure and reverse the order of the steps while using opposite operations to obtain the solution of the problem. Therefore, P is equal to NP; and there are no NP problems. Furthermore, all NP problems are P problems, according to the examples in this paper.

**Category:** Combinatorics and Graph Theory

[66] **viXra:1908.0480 [pdf]**
*replaced on 2019-08-27 15:56:18*

**Authors:** A. A. Frempong

**Comments:** 10 Pages. Copyright © by A. A. Frempong

By applying differential and integral calculus, this paper covers the principles and procedures for producing the solution of a problem, given the procedure for checking the correctness of the solution of a problem, and vice versa. If one is able to check quickly and completely, the correctness of the solution of a problem, one should also be able to produce the solution of the problem by reversing the order of the steps of the checking process, while using opposite operations in each step. The above principles were applied to four examples from calculus as well as to an example from geometry. Even though in calculus, one normally uses differentiation to check the correctness of an integration result, one will differentiate a function first, and then integrate the derivative to obtain the original function. One will differentiate the trigonometric functions, tan x, cot x, sec x and csc x; followed by integrating each derivative to obtain each original function. The results show that the solution process and the checking process are inverses of each other. In checking the correctness of the solution of a problem, one should produce the complete checking procedure which includes the beginning, the middle, and the end of the problem. Checking only the correctness of the final answer or statement is incomplete checking. To facilitate complete checking, the question should always be posed such that one is compelled to show a complete checking procedure from which the solution procedure can be produced. A general application of P = NP is that, if the correctness of the solution of a problem can be checked quickly and it is difficult to write a solution procedure, then first, write a complete checking procedure and reverse the order of the steps while using opposite operations to obtain the solution of the problem. Therefore, P is equal to NP.

**Category:** Combinatorics and Graph Theory

[65] **viXra:1908.0480 [pdf]**
*replaced on 2019-08-25 00:49:40*

**Authors:** A. A. Frempong

**Comments:** 10 Pages. Copyright © by A. A. Frempong

By applying differential and integral calculus, this paper covers the principles and procedures for producing the solution of a problem, given the procedure for checking the correctness of the solution of a problem, and vice versa. If one is able to check quickly and completely, the correctness of the solution of a problem, one should also be able to produce the solution of the problem by reversing the order of the steps of the checking process, while using opposite operations in each step. The above principles were applied to four examples from calculus as well as to an example from geometry. Even though in calculus, one normally uses differentiation to check the correctness of an integration result, one will differentiate a function first, and then integrate the derivative to obtain the original function. One will differentiate the trigonometric functions, tan x, cot x, sec x and csc x; followed by integrating each derivative to obtain each original function. The results show that the solution process and the checking process are inverses of each other. In checking the correctness of the solution of a problem, one should produce the complete checking procedure which includes the beginning, the middle, and the end of the problem. Checking only the correctness of the final answer or statement is incomplete checking. To facilitate complete checking, the question should always be posed such that one is compelled to show a complete checking procedure from which the solution procedure can be produced. A general application of P = NP is that, if the correctness of the solution of a problem can be checked quickly and it is difficult to write a solution procedure, then first, write a complete checking procedure and reverse the order of the steps while using opposite operations to obtain the solution of the problem. Therefore, P is equal to NP.

**Category:** Combinatorics and Graph Theory

[64] **viXra:1908.0355 [pdf]**
*replaced on 2019-08-30 08:45:46*

**Authors:** Anna Knezevic, Greg Cohen, Marina Domanskaya

**Comments:** 85 pages; this research was partly supported by the School of Electrical Engineering, Computing and Mathematical Sciences of the Curtin University (Australia) whose member is one of the authors (Anna Knezevic)

The permanent’s polynomial-time computability over fields of characteristic 3 for k-semiunitary matrices (i.e. n×n-matrices A such that ��������(����^�� − ��_��) = ��) in the case k ≤ 1 and its #3P-completeness for any k > 1 (Ref. 9) is a result that essentially widens our understanding of the computational complexity boundaries for the permanent modulo 3. Now we extend this result to study more closely the case k > 1 regarding the (n-k)×(n-k)sub-permanents (or permanent-minors) of a unitary n×n-matrix and their possible relations, because an (n-k)×(n-k)-submatrix of a unitary n×n-matrix is generically a ksemi-unitary (n-k)×(n-k)-matrix. The following paper offers a way to receive a variety of such equations of different sorts, in the meantime extending (in its second chapter divided into subchapters) this direction of research to reviewing all the set of polynomial-time permanent-preserving reductions and equations for a generic matrix’s sub-permanents they might yield, including a number of generalizations and formulae (valid in an arbitrary prime characteristic) analogical to the classical identities relating the minors of a matrix and its inverse. Moreover, the second chapter also deals with the Hamiltonian cycle polynomial in characteristic 2 that surprisingly demonstrates quite a number of properties very similar to the corresponding ones of the permanent in characteristic 3. Besides, the paper’s third chapter is devoted to the computational complexity issues of the permanent and some related functions on a variety of Cauchy matrices and their certain generalizations, including constructing a polynomial-time algorithm (based on them) for the permanent of an arbitrary square matrix in characteristic 5 and conjecturing the existence of a similar scheme in characteristic 3. Throughout the paper, we investigate various matrix compressions and transformations preserving the permanent and related functions in certain finite characteristics. And, as an auxiliary algebraic tool supposed for an application when needed in all the constructions we’re going to discuss in the present article, we’ll introduce and utilize a special principle involving a field’s extension by a formal infinitesimal and allowing, provided a number of conditions are fulfilled, to reduce the computation of a polynomial over a field to solving a system of algebraic equations in polynomial time

**Category:** Combinatorics and Graph Theory

[63] **viXra:1908.0355 [pdf]**
*replaced on 2019-08-30 02:58:42*

**Authors:** Anna Knezevic, Greg Cohen, Marina Domanskaya

**Comments:** 85 Pages. this research was partly supported by the School of Electrical Engineering, Computing and Mathematical Sciences of the Curtin University (Australia) whose member is one of the authors (Anna Knezevic)

The permanent’s polynomial-time computability over fields of characteristic 3 for k-semiunitary matrices (i.e. n×n-matrices A such that ��������(����^�� − ��_��) = ��) in the case k ≤ 1 and its #3P-completeness for any k > 1 (Ref. 9) is a result that essentially widens our understanding of the computational complexity boundaries for the permanent modulo 3. Now we extend this result to study more closely the case k > 1 regarding the (n-k)×(n-k)sub-permanents (or permanent-minors) of a unitary n×n-matrix and their possible relations, because an (n-k)×(n-k)-submatrix of a unitary n×n-matrix is generically a ksemi-unitary (n-k)×(n-k)-matrix.
The following paper offers a way to receive a variety of such equations of different sorts, in the meantime extending (in its second chapter divided into subchapters) this direction of research to reviewing all the set of polynomial-time permanent-preserving reductions and equations for a generic matrix’s sub-permanents they might yield, including a number of generalizations and formulae (valid in an arbitrary prime characteristic) analogical to the classical identities relating the minors of a matrix and its inverse. Moreover, the second chapter also deals with the Hamiltonian cycle polynomial in characteristic 2 that surprisingly demonstrates quite a number of properties very similar to the corresponding ones of the permanent in characteristic 3.
Besides, the paper’s third chapter is devoted to the computational complexity issues of the permanent and some related functions on a variety of Cauchy matrices and their certain generalizations, including constructing a polynomial-time algorithm (based on
them) for the permanent of an arbitrary square matrix in characteristic 5 and conjecturing the existence of a similar scheme in characteristic 3.
Throughout the paper, we investigate various matrix compressions and transformations preserving the permanent and related functions in certain finite characteristics. And, as an auxiliary algebraic tool supposed for an application when needed in all the constructions we’re going to discuss in the present article, we’ll introduce and utilize a special principle involving a field’s extension by a formal infinitesimal and allowing, provided a number of conditions are fulfilled, to reduce the computation of a polynomial over a field to solving a system of algebraic equations in polynomial time

**Category:** Combinatorics and Graph Theory

[62] **viXra:1906.0501 [pdf]**
*replaced on 2019-07-22 12:27:42*

**Authors:** Marco Ripà

**Comments:** 12 Pages.

In this paper, we show enhanced upper bounds of the nontrivial n_1 × n_2 × n_3 points problem for every n_1 ≤ n_2 ≤ n_3 < 6. We present new patterns that drastically improve the previously known algorithms for finding minimum-link covering paths, solving completely a few cases (e.g., n_1 = n_2 = 3 and n_3 = 4).

**Category:** Combinatorics and Graph Theory

[61] **viXra:1906.0501 [pdf]**
*replaced on 2019-07-09 12:05:25*

**Authors:** Marco Ripà

**Comments:** 12 Pages. This paper has not been submitted before to any peer-reviewed journal © Marco Ripà

In this paper, we show enhanced upper bounds of the nontrivial n_1 × n_2 × n_3 points problem for every n_1 ≤ n_2 ≤ n_3 < 6. We present new patterns that drastically improve the previously known algorithms for finding minimum-link covering paths, solving completely a few cases (e.g., n_1 = n_2 = 3 and n_3 = 4).

**Category:** Combinatorics and Graph Theory

[60] **viXra:1906.0501 [pdf]**
*replaced on 2019-07-08 21:41:41*

**Authors:** Marco Ripà

**Comments:** 12 Pages. This paper has not been submitted before to any peer-reviewed journal © Marco Ripà

**Category:** Combinatorics and Graph Theory

[59] **viXra:1906.0501 [pdf]**
*replaced on 2019-07-07 19:40:18*

**Authors:** Marco Ripà

**Comments:** 12 Pages.

**Category:** Combinatorics and Graph Theory

[58] **viXra:1903.0256 [pdf]**
*replaced on 2019-04-25 16:14:48*

**Authors:** Roman Galay

**Comments:** 12 Pages. The following short article offers a couple of algebraically entangled polynomial-time algorithms for the graph isomorphism and clique problems whose correctness is yet to be determined either empirically or through attempting to find proofs.

As it follows from Gödel's incompleteness theorems, any consistent formal system of axioms and rules of inference should imply a true unprovable statement. Actually this fundamental principle can be efficiently applicable in Computational Mathematics and Complexity Theory concerning the computational complexity of problems from the class NP, particularly and especially the NP-complete ones. While there is a wide set of algorithms for these problems that we call heuristic, the correctness or/and complexity of each concrete algorithm (or the probability of its correct and polynomial-time work) on a class of instances is often too difficult to determine, although we may also assume the existence of a variety of algorithms for NP-complete problems that are both correct and polynomial-time on all the instances from a given class (where the given problem remains NP-complete), but whose correctness or/and polynomial-time complexity on the class is impossible to prove as an example for Gödel's theorems. However, supposedly such algorithms should possess a certain complicatedness of processing the input data and treat it in a certain algebraically “entangled” manner. The same algorithmic analysis in fact concerns all the other significant problems and subclasses of NP, such as the graph isomorphism problem and its associated complexity class GI.
The following short article offers a couple of algebraically entangled polynomial-time algorithms for the graph isomorphism and clique problems whose correctness is yet to be determined either empirically or through attempting to find proofs.

**Category:** Combinatorics and Graph Theory

[57] **viXra:1903.0256 [pdf]**
*replaced on 2019-03-20 11:58:28*

**Authors:** Roman Galay

**Comments:** 8 Pages. The following short article offers a couple of algebraically "entangled" polynomial-time algorithms for the graph isomorphism and clique problems whose correctness is yet to be determined either empirically or through attempting to find proofs.

As it follows from Gödel's incompleteness theorems, any consistent formal system of axioms and rules of inference should imply a true unprovable statement. Actually this fundamental principle can be efficiently applicable in Computational Mathematics and Complexity Theory concerning the computational complexity of problems from the class NP, particularly and especially the NP-complete ones. While there is a wide set of algorithms for these problems that we call heuristic, the correctness or/and complexity of each concrete algorithm (or the probability of its correct and polynomial-time work) on a class of instances is often too difficult to determine, although we may also assume the existence of a variety of algorithms for NP-complete problems that are both correct and polynomial-time on all the instances from a given class (where the given problem remains NP-complete), but whose correctness or/and polynomial-time complexity on the class is impossible to prove as an example for Gödel's theorems. However, supposedly such algorithms should possess a certain complicatedness of processing the input data and treat it in a certain algebraically “entangled” manner. The same algorithmic analysis in fact concerns all the other significant problems and subclasses of NP, such as the graph isomorphism problem and its associated complexity class GI.
The following short article offers a couple of algebraically entangled polynomial-time algorithms for the graph isomorphism and clique problems whose correctness is yet to be determined either empirically or through attempting to find proofs.

**Category:** Combinatorics and Graph Theory

[56] **viXra:1807.0384 [pdf]**
*replaced on 2018-07-25 16:31:37*

**Authors:** Marco Ripà, Valerio Bencini

**Comments:** 14 Pages.

In this paper we describe two new patterns, in order to improve the upper bound for the Ripà’s n X n X n points problem, saving a few lines for many values of n. The new upper bound, for any n≥6, becomes h_u(n)=int((3/2*n^2)+int(n/4)-int((n-1)/4)+int((n+1)/4)-int((n+2)/4)+n-2, where int(x)≔floor(x).

**Category:** Combinatorics and Graph Theory

[55] **viXra:1806.0045 [pdf]**
*replaced on 2018-06-08 08:27:08*

**Authors:** Thinh D. Nguyen

**Comments:** 4 Pages.

In this note, we show that graph isomorphism and some of its variant are both reducible to circuit isomorphism problem, in polynomial time.

**Category:** Combinatorics and Graph Theory

[54] **viXra:1805.0377 [pdf]**
*replaced on 2018-05-26 14:50:45*

**Authors:** Prashanth R. Rao

**Comments:** 2 Pages.

In this paper we show a method to encode graphs with a numerical value that follows unique labeling of each vertex or node and unique labeling of each edge of a graph with unique prime numbers. Each edge is defined as the connectivity between two vertices, therefore two vertices or nodes connected by an edge may be represented by the “ edge-nodes value ” derived by raising the prime number representing the edge to the product of the primes representing the two nodes that are connected by that edge. Multiplying all the “edge-nodes values” of a single graph will represent a unique number albeit very large in majority of cases. Given this unique number called the “Edge-nodes values product”, it is possible to derive the structure of the given graph. This encoding may allow new approaches to graph isomorphism, cryptography, quantum computing, data security, artificial intelligence, etc.

**Category:** Combinatorics and Graph Theory

[53] **viXra:1803.0163 [pdf]**
*replaced on 2018-03-15 04:32:21*

**Authors:** Juan Moreno Borrallo

**Comments:** 4 Pages.

In this paper it is discussed the following problem: "A mathematician wants to divide is garden into consecutive prime parts (first in two parts, after in three parts, and so on), only making straight paths, in a simple way (without retracing his own steps), and without going out of his plot of land. In how many parts can the mathematician divide his garden?"

**Category:** Combinatorics and Graph Theory

[52] **viXra:1711.0432 [pdf]**
*replaced on 2018-01-28 13:19:20*

**Authors:** Arsen A. Movsesyan

**Comments:** 23 Pages.

In the article, the Gauss’s problem on the number of integer points for a circle and a ball in the framework of an integer lattice is reformulated in an equivalent way and reduces to solving two combinatorial tasks for a circular and spherical "layer" in the framework of Quantum Discrete Space. These tasks are solved using trigonometric functions defined on a set of integers whose range of values is also integers, and other new mathematical tools. It comes not about evaluative solutions, but about exact solutions, which, if necessary, can be transferred to a circle and a ball. In doing so not only specific formulas for determine the exact number of solutions are presented, but also the formulas for enumerating the corresponding pairs and triples of integers. The importance of obtained solutions lies in the fact that they determine the analytical likenesses of not only the circumference and the sphere in the Quantum Discrete Space, but also point to the possibility of constructing of the likenesses of ellipse, cone, hyperboloid and other figures.

**Category:** Combinatorics and Graph Theory

[51] **viXra:1709.0242 [pdf]**
*replaced on 2017-09-17 03:01:55*

**Authors:** Ikhlef Bechar

**Comments:** 50 Pages.

This paper is concerned with the problem of exact MAP inference in general higher-order
graphical models by means of a traditional linear programming relaxation approach. In fact, the proof that we have developed in this paper is a rather simple algebraic proof being
made straightforward, above all, by the introduction of two novel algebraic tools. Indeed, on the one hand, we introduce the notion of delta-distribution which merely stands for the difference of two arbitrary probability distributions, and which mainly serves to alleviate the sign constraint inherent to a traditional probability distribution. On the other hand, we develop an approximation framework of general discrete functions by means of an orthogonal projection expressing in terms of linear combinations of function margins with respect to a given collection of point subsets, though, we rather exploit the latter approach for the purpose of modeling locally consistent sets of discrete functions from a global perspective. After that, as a first step, we develop from scratch the expectation optimization framework which is nothing else than a reformulation, on stochastic grounds, of the convex-hull approach, as a second step, we develop the traditional LP relaxation of such an expectation optimization approach, and we
show that it enables to solve the MAP inference problem in graphical models under rather general assumptions. Last but not least, we describe an algorithm which allows to compute an exact MAP solution from a perhaps fractional optimal (probability) solution of the proposed LP relaxation.

**Category:** Combinatorics and Graph Theory

[50] **viXra:1708.0163 [pdf]**
*replaced on 2017-08-16 02:10:47*

**Authors:** A. A. Frempong

**Comments:** 12 Pages. Copyright © by A. A. Frempong

The traveling salesman can determine by hand, with zero or negligible error, the shortest route from home base city to visit once, each of three cities, 10 cities, 20 cities, 100 cities, or 1000 cities, and return to the home base city. The general approach to solving the different types of NP problems is the same, except that sometimes, specific techniques may differ from each other according to the process involved in the problem. In the salesman problem, the first step is to arrange the data in the problem in increasing order, since one's interest is in the shortest distances. The main principle here is that the shortest route is the sum of the shortest distances such that the salesman visits each city once and returns to the starting city. The approach in this paper is different from the author's previous approach (viXra:1505.0167) in which the needed distances not among the least ten distances were added to the least ten distances before route construction began. In this paper, one starts with only the least ten distances and only if a needed distance is not among the set of the least ten distances, would one consider distances greater than those in the set of the ten least distances.
The shortest route to visit nine cities and return to the starting city was found in this paper. It was also found out that even though the length of the shortest route is unique, the sequence of the cities involved is not unique. The approach used in this paper can be applied in workforce project management and hiring, as well as in a country's workforce needs and immigration quota determination. Since an approach that solves one of these problems can also solve other NP problems, and the traveling salesman problem has been solved, all NP problems can be solved, provided that one has an open mind and continues to think. If all NP problems can be solved, then all NP problems are P problems and therefore, P is equal to NP. The CMI Millennium Prize requirements have been satisfied

**Category:** Combinatorics and Graph Theory

[49] **viXra:1701.0341 [pdf]**
*replaced on 2017-02-02 01:39:37*

**Authors:** Alexey V. Komkov

**Comments:** 2 Pages.

This work contains certificates numbers Van der Waerden, was found using SAT Solver. These certificates establish the best currently known lower bounds of the numbers Van der Waerden W( 7, 3 ), W( 8, 3 ), W( 9, 3 ), W( 10, 3 ), W( 11, 3 ).

**Category:** Combinatorics and Graph Theory

[48] **viXra:1611.0385 [pdf]**
*replaced on 2018-10-30 05:38:20*

**Authors:** Alexey V. Komkov

**Comments:** 114 Pages.

Genetic algorithm is a good tool for finding the global minimum in many discrete problems. In particular, it has proven itself in problems where there is no any apriori information about the possibilities of narrowing the search, or the specifics of the problem do not involve such. This work describes the procedure of using a genetic algorithm as applied to the search of van der Waerden numbers. Some new lower bounds of van der Waerden numbers were found using this procedure. Using the genetic algorithm, the author has found the best lower estimates for the Van der Waerden number W (7, 4), W (5, 5), W (6, 5), W (5, 6).

**Category:** Combinatorics and Graph Theory

[47] **viXra:1608.0250 [pdf]**
*replaced on 2018-05-28 01:43:16*

**Authors:** Atul Mehta

**Comments:** 7 Pages.

In this paper, we explore the connections between graphs and Turing
machines. A method to construct Turing machines from a general
undirected graph is provided. Determining whether a Hamiltonian
cycle exists is now shown to be equivalent to solving the halting
problem.
We investigate applications of the halting problem to problems in number theory.
A modified version of the classical Turing machine is now developed to
solve certain classes of computational problems.

**Category:** Combinatorics and Graph Theory

[46] **viXra:1607.0132 [pdf]**
*replaced on 2017-02-21 00:02:16*

**Authors:** Ihsan Raja Muda Nasution

**Comments:** 2 Pages.

In this paper, we solve the four-color problem by a new algorithm.

**Category:** Combinatorics and Graph Theory