**Previous months:**

2009 - 0908(1)

2010 - 1001(3) - 1003(4) - 1004(1) - 1006(1) - 1009(1) - 1010(1)

2011 - 1101(1) - 1103(1) - 1106(1)

2012 - 1202(2) - 1204(1) - 1208(2) - 1209(2) - 1210(3) - 1211(2)

2013 - 1303(1) - 1304(2) - 1305(2) - 1307(5) - 1308(2) - 1309(7) - 1310(2) - 1311(2)

2014 - 1401(1) - 1403(1) - 1404(2) - 1408(2) - 1409(3) - 1411(6) - 1412(1)

2015 - 1501(1) - 1502(1) - 1503(4) - 1504(2) - 1505(2) - 1507(1) - 1508(3) - 1509(2) - 1510(1) - 1511(3) - 1512(3)

2016 - 1601(3) - 1602(6) - 1603(2) - 1604(8) - 1605(2) - 1606(1) - 1608(1)

Any replacements are listed further down

[110] **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

[109] **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

[108] **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

[107] **viXra:1605.0023 [pdf]**
*submitted on 2016-05-03 01:14:32*

**Authors:** Marius Coman

**Comments:** 151 Pages.

The purpose of this book is to show that the method of concatenation can be a powerful tool in
number theory and, in particular, in obtaining possible infinite sequences of primes.

**Category:** Combinatorics and Graph Theory

[106] **viXra:1604.0188 [pdf]**
*submitted on 2016-04-12 01:22:25*

**Authors:** Linfan Mao

**Comments:** 141 Pages.

The International J.Mathematical Combinatorics (ISSN 1937-1055) is a fully refereed international journal, sponsored by the MADIS of Chinese Academy of Sciences and published in USA quarterly comprising 100-150 pages approx. per volume, which publishes original researchpapers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences.

**Category:** Combinatorics and Graph Theory

[105] **viXra:1604.0187 [pdf]**
*submitted on 2016-04-12 01:23:59*

**Authors:** B. Chaluvaraju, H.S. Boregowda, S. Kumbinarsaiah

**Comments:** 6 Pages.

The binding number of a graph G = (V,E) is deﬁned to be the minimum of |N(X)|/|X| taken over all nonempty set X ⊆ V (G) such that N(X) 6= V (G). In this article, we explore the properties and bounds on binding number of some special classes of trees.

**Category:** Combinatorics and Graph Theory

[104] **viXra:1604.0185 [pdf]**
*submitted on 2016-04-12 01:26:19*

**Authors:** Suleyman Senyurt, Abdussamet Calıskan, Unzile Celik

**Comments:** 7 Pages.

In this paper, let (α,α∗) be Bertrand curve pair, when the unit Darboux vector of the α∗ curve are taken as the position vectors, the curvature and the torsion of Smarandache curve are calculated. These values are expressed depending upon the α curve. Besides, we illustrate example of our main results.

**Category:** Combinatorics and Graph Theory

[103] **viXra:1604.0184 [pdf]**
*submitted on 2016-04-12 01:27:49*

**Authors:** Nutan G. Nayak

**Comments:** 8 Pages.

In this paper, we obtained the characterization of net-regular signed graphs and also established the spectrum for one class of heterogeneous unbalanced net-regular signed complete graphs.

**Category:** Combinatorics and Graph Theory

[102] **viXra:1604.0183 [pdf]**
*submitted on 2016-04-12 01:29:16*

**Authors:** R. Ponraj, M. Maria Adaickalam, R. Kala

**Comments:** 8 Pages.

A graph with a quotient cordial labeling is called a quotient cordial graph. We investigate the quotient cordial labeling behavior of path, cycle, complete graph, star, bistar etc.

**Category:** Combinatorics and Graph Theory

[101] **viXra:1604.0182 [pdf]**
*submitted on 2016-04-12 01:30:56*

**Authors:** Linfan MAO

**Comments:** 141 Pages.

The Mathematical Combinatorics (International Book Series) is a fully refereed international book series with ISBN number on each issue, sponsored by the MADIS of Chinese Academy of Sciences and published in USA quarterly comprising 100-150 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences.

**Category:** Combinatorics and Graph Theory

[100] **viXra:1604.0111 [pdf]**
*submitted on 2016-04-05 23:26:53*

**Authors:** A. A. Frempong

**Comments:** 8 Pages. Copyright © A.A. Frempong. Reference: P vs NP:Solutions of NP Problems, viXra:1408.0204 by A. A. Frempong

This paper covers the principles and procedures for producing the solution of a problem given the procedure for checking the solution of the problem and vice versa. If a problem can be checked in polynomial time, it can be solved in polynomial time, provided a complete checking procedure is available. From a point A, if one uses one's feet to measure a certain distance by counting steps forwards to a point B, and one wants to check the correctness of the measurement, one would count backwards from the point B using one's feet to see if one returns to exactly the point A. If one returns to A, the forward counting is correct, otherwise it is incorrect. If one counted backwards first from the point B to the point A, one could also count forwards from A to B. Before computers were used in filing taxes in the United States, when one prepared a tax return and wanted to check for arithmetic errors, one would reverse the arithmetic steps from the last arithmetic statement backwards all the way to the first entry on the tax form; and if one obtains a zero after reversing the steps, one was sure that there were no arithmetic errors on the tax form (That is, one began with zero entry going forward and one returned with a zero entry). So also, if one is able to check quickly the correctness of the solution to a problem, one should also be able to produce the solution of the problem by reversing the steps of the checking process while using opposite operations in each step. If a complete checking process is available, the solution process can be obtained by reversing the steps of the checking while using opposite operations in each step. In checking the correctness of the solution to a problem, one should produce the complete checking process which includes the end of the problem and the beginning of the problem. Checking only the final answer or statement is incomplete checking. Since the solution process and the checking process are inverses of each other, knowing one of them, one can obtain the other by reversing the steps while using opposite operations. To facilitate complete checking, the question should always be posed such that one is compelled to show a checking procedure from which the solution procedure can be deduced. Therefore P is always equal to NP.

**Category:** Combinatorics and Graph Theory

[99] **viXra:1604.0012 [pdf]**
*submitted on 2016-04-02 01:31:57*

**Authors:** Ton Kloks, Jan van Leeuwen, Fu-Hong Liu, Hsiang-Hsuan Liu, Richard B. Tan, Yue-Li Wang

**Comments:** 12 Pages.

We show that the geodetic game is decidable in polynomial time for various classes of AT-free graphs.

**Category:** Combinatorics and Graph Theory

[98] **viXra:1603.0218 [pdf]**
*submitted on 2016-03-15 07:27:51*

**Authors:** Ali Reza Najar Saligheh

**Comments:** 6 Pages. English and French languages.

First we will prove that Kk (the complete graph with k vertices) is a minor of every graph with chromatic number k. Then we will prove some other statements such as the four color theorem and the Hadwiger conjecture. We will not use computer-assisted proofs.

**Category:** Combinatorics and Graph Theory

[97] **viXra:1603.0041 [pdf]**
*submitted on 2016-03-03 10:06:19*

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

**Comments:** 19 Pages. http://www.newtheory.org

In this article, we combine the concept of bipolar neutrosophic set and graph theory. We introduce the notions of bipolar single valued neutrosophic graphs, strong bipolar single valued neutrosophic graphs, complete bipolar single valued neutrosophic graphs, regular bipolar single valued neutrosophic graphs and investigate some of their related properties

**Category:** Combinatorics and Graph Theory

[96] **viXra:1602.0157 [pdf]**
*submitted on 2016-02-13 11:02:28*

**Authors:** Mohamed Eleiche

**Comments:** 12 Pages.

The minimum-travel-cost algorithm is a dynamic programming algorithm to compute an exact and deterministic lower bound for the general case of the traveling salesman problem (TSP). The algorithm is presented with its mathematical proof and asymptotic analysis. It has a (n2) complexity. A program is developed for the implementation of the algorithm and successfully tested among well known TSP problems.

**Category:** Combinatorics and Graph Theory

[95] **viXra:1602.0153 [pdf]**
*submitted on 2016-02-13 09:00:10*

**Authors:** Florentin Smarandache

**Comments:** 4 Pages.

The science's function is realizing the natural world, developing our society in coordination with natural laws and mathematics provides the quantitative tool and method for solving problems helping with that understanding. Generally, understanding a natural thing by mathematical ways or means to other sciences are respectively establishing mathematical model on typical characters of it with analysis first, and then forecasting its behaviors, and finally, directing human beings for hold on its essence by that model.

**Category:** Combinatorics and Graph Theory

[94] **viXra:1602.0121 [pdf]**
*submitted on 2016-02-10 11:59:09*

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

**Comments:** 33 pages

The notion of interval valued neutrosophic sets is a generalization of fuzzy sets, intuitionistic fuzzy sets, interval valued fuzzy sets, interval valued intuitionstic fuzzy sets and single valued neutrosophic sets. We apply for the first time the concept of interval valued neutrosophic sets, an instance of neutrosophic sets, to graph theory. We introduce certain types of interval valued neutrosophc graphs (IVNG) and investigate some of their properties with proofs and examples.

**Category:** Combinatorics and Graph Theory

[93] **viXra:1602.0120 [pdf]**
*submitted on 2016-02-10 12:02:44*

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

**Comments:** 21 pages

In this paper, we discuss a subclass of interval valued neutrosophic graphs called strong interval valued neutrosophic graphs which were introduced by Broumi et al [41]. The operations of Cartesian product, composition, union and join of two strong interval valued neutrosophic graphs are defined. Some propositions involving strong interval valued neutrosophic graphs are stated and proved.

**Category:** Combinatorics and Graph Theory

[92] **viXra:1602.0117 [pdf]**
*submitted on 2016-02-10 06:26:54*

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

**Comments:** 16 pages

The notion of single valued neutrosophic sets is a generalization of fuzzy sets, intuitionistic fuzzy sets. We apply the concept of single valued neutrosophic sets, an instance of neutrosophic sets, to graphs. We introduce certain types of single valued neutrosophic graphs (SVNG) and investigate some of their properties with proofs and examples.

**Category:** Combinatorics and Graph Theory

[91] **viXra:1602.0076 [pdf]**
*submitted on 2016-02-06 11:00:18*

**Authors:** Many Authors

**Comments:** 145 Pages.

Collection of papers on combinatorics.

**Category:** Combinatorics and Graph Theory

[90] **viXra:1601.0247 [pdf]**
*submitted on 2016-01-22 21:57:43*

**Authors:** Quang Nguyen Van

**Comments:** 14 pages. We have other one written in tex.file

We choose one of four colors as a temporary color for all regions that have not been colored,and made the initial conditions corresponding to The Four Color Theorem. If these conditions hold for any n regions figure, then they will hold for n + 1 regions figure - formed n regions figure by adding next region. By induction, step by step we have proved The Four Color theorem successfully on paper (in 2015).

**Category:** Combinatorics and Graph Theory

[89] **viXra:1601.0200 [pdf]**
*submitted on 2016-01-18 09:45:28*

**Authors:** Prashanth R. Rao

**Comments:** 3 Pages.

Abstract: In a previous paper we described a method to represent graph information as a single numerical value by distinctly labeling each of its vertices with unique primes. In this paper, we modify the previous approach to again represent a graph as a single numeric value, we log transform this value and approximate it with an optimum value which if minimized by appropriate prime labeling of the graph should allow us to compare it with another graph on which an identical algorithm is implemented. Identical optimum value minima may be expected to indicate graph isomorphism.

**Category:** Combinatorics and Graph Theory

[88] **viXra:1601.0048 [pdf]**
*submitted on 2016-01-06 05:30:54*

**Authors:** Quang Nguyen Van

**Comments:** 5 Pages.

We introduce an efficient method for exporting a sequence of prime numbers by using Excel.

**Category:** Combinatorics and Graph Theory

[87] **viXra:1512.0343 [pdf]**
*submitted on 2015-12-17 01:55:47*

**Authors:** Mikhail Kovalyov

**Comments:** 8 Pages.

The paper provides an intuitive and very short derivation of the normal distribution and the Stirling and Wallis formulas.

**Category:** Combinatorics and Graph Theory

[86] **viXra:1512.0322 [pdf]**
*submitted on 2015-12-15 03:00:14*

**Authors:** Dhananjay P. Mehendale

**Comments:** 6 pages

In this paper we develop a novel characterization for isomorphism of graphs. The characterization is obtained in terms of ordered adjacency lists to be associated with two given labeled graphs. We show that the two given labeled graphs are isomorphic if and only if their associated ordered adjacency lists can be made identical by applying the action of suitable transpositions on any one of these lists. We discuss in brief the complexity of the algorithm for deciding isomorphism of graphs and show that it is of the order of the cube of number of the number of edges.

**Category:** Combinatorics and Graph Theory

[85] **viXra:1512.0222 [pdf]**
*submitted on 2015-12-04 15:55:12*

**Authors:** Prashanth R. Rao

**Comments:** 2 Pages.

We present a simple theoretical strategy to represent using a single numerical
value “A” called the prime vertex labeling Adjacency value, all structural information
encoded in a graph. This strategy has the potential to allow us to reconstruct the graph in
its entirety based on a single number. To do so we assume that we have access to a large
list of prime numbers which are infinite in number. This method will allow us to store
graph backbone as a numerical value for retrieval and re-use and may also allow
development of algorithms that exploit this representation feature as shortcut to address
graph isomorphism.

**Category:** Combinatorics and Graph Theory

[84] **viXra:1511.0225 [pdf]**
*submitted on 2015-11-23 15:08:55*

**Authors:** Richard J. Mathar

**Comments:** 31 Pages. Includes a Java program licensed under the LGPL v3.0.

A terrace form assigns an integer altitude to each point of a finite two-dimensional square grid
such that the maximum altitude difference between a point and its four neighbors is one.
It is 2-way monotonic if the sign of this altitude difference is zero or one for steps to the East or steps
to the South. We provide tables for the number of 2-way monotonic terrace forms as a function of grid size
and maximum altitude difference, and point at the equivalence to the number of 3-colorings of the grid.

**Category:** Combinatorics and Graph Theory

[83] **viXra:1511.0167 [pdf]**
*submitted on 2015-11-19 06:14:36*

**Authors:** Ali Reza Najar Saligheh

**Comments:** 11 Pages. English and French languages.

In order to prove the Four color theorem without using computer, I will prove that no disproof can exist. I will look for some characteristics needed for a disproof and then I will prove that these characteristics can not exist.

**Category:** Combinatorics and Graph Theory

[82] **viXra:1511.0015 [pdf]**
*submitted on 2015-11-02 15:32:57*

**Authors:** Richard J. Mathar

**Comments:** Pages 9 to 21 are a JAVA program distributed under the LGPL v3.

The multinomial coefficients count the number of ways (of permutations) of
placing a number of partially distinguishable objects on a line, taking ordering
into account. A well-known two-parametric family of counts arises if there are objects
of c distinguishable colors and m objects of each color, m*c objects in total, to be placed on line.
In this work we propose an algorithm to count the permutations where
no two objects of the same color appear side-by-side on the line. This eliminates
all permutations with "clusters" of colors. Essentially we represent filling
the line sequentially with objects as a tree of states where each node
matches one partially filled line. Subtrees are merged if they have the same
branching structure, and weights are assigned to nodes in the tree keeping track
of how many mergers take place. This is implemented in a JAVA program; numerical results
confirm Hardin's earlier counts for this kind of restricted permutations.

**Category:** Combinatorics and Graph Theory

[81] **viXra:1510.0404 [pdf]**
*submitted on 2015-10-27 03:05:17*

**Authors:** Dainis Zeps

**Comments:** 16 Pages. The article is a Mathematica notebook

In this article we explore 4-critical graphs using Mathematica. We generate graph patterns according [1]. Using the base graph, minimal planar multiwheel and in the same time minimal according projective pattern built multiwheel, we build minimal multiwheels according [1], We forward two conjectures according graphs augmented according considered patterns and their 4-criticallity, and argue them to be proved here if the paradigmatic examples of this article are accepted to be parts of proofs.

**Category:** Combinatorics and Graph Theory

[80] **viXra:1509.0150 [pdf]**
*submitted on 2015-09-17 10:04:18*

**Authors:** Marco Ripà

**Comments:** 5 Pages.

The classic thinking problem, the “Nine Dots Puzzle”, is widely used in courses on creativity and appears in a lot of games magazines. Here are two mutually exclusive conjectures about the generic solution of the problem of the 3k dots spread to 3 X 3 X … X 3 points, in a k-dimensional space.

**Category:** Combinatorics and Graph Theory

[79] **viXra:1509.0140 [pdf]**
*submitted on 2015-09-16 14:19:53*

**Authors:** Richard J. Mathar

**Comments:** 32 Pages. Computer program distributed under the LGPLv3 license .

We provide a C++ program which searches for the smallest number of pouring steps
that convert a set of jugs with fixed (integer) capacities and some initial known (integer)
water contents
into another state with some other prescribed water contents. Each step
requires to pour one jug into another
without spilling
until either the source jug is empty or the
drain jug is full-because the model assumes the jugs have irregular shape and no
further marks.
The program simply
places the initial jug configuration at the root of the tree
of state diagrams and deploys the branches (avoiding loops) recursively by
generating all possible states from known states in one pouring step.

**Category:** Combinatorics and Graph Theory

[78] **viXra:1508.0201 [pdf]**
*submitted on 2015-08-24 19:58:13*

**Authors:** Marco Ripà

**Comments:** Pages.

We provide an optimal strategy to solve the n X n X n points problem inside the box, considering only 90° and 45° 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 n3 dots, for any n > 5. 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 h_u(n) − h_l(n) between the upper and the lower bound, proving that it is ≤ 0.5 ∙ n ∙ (n + 3), if n > 1.

**Category:** Combinatorics and Graph Theory

[77] **viXra:1508.0085 [pdf]**
*submitted on 2015-08-11 11:37:07*

**Authors:** Philip Gibbs

**Comments:** 16 Pages.

The Ulam numbers form an increasing sequence beginning 1,2 such that each subsequent number can be uniquely represented as the sum of two smaller Ulam numbers. An algorithm is described and implemented in Java to compute the first billion Ulam numbers.

**Category:** Combinatorics and Graph Theory

[76] **viXra:1508.0045 [pdf]**
*submitted on 2015-08-05 07:59:00*

**Authors:** Philip Gibbs

**Comments:** 2 Pages.

A conjecture on the quasi-periodic behaviour of Ulam sequences

**Category:** Combinatorics and Graph Theory

[75] **viXra:1507.0124 [pdf]**
*submitted on 2015-07-16 09:22:03*

**Authors:** editor Linfan Mao

**Comments:** 152 Pages.

There are 11 papers in this volume.
Paper 1: Mathematics After CC Conjecture - Combinatorial Notions and Achievements, is a report of mine on the International Conference on Combinatorics, Graph Theory, Topology and Geometry, January 29-31, 2015, Shanghai, P. R. China, including Smarandache systems, Smarandache geometries.
Paper 2: Timelike-Spacelike Mannheim Pair Curves Spherical Indicators Geodesic Curvatures and Natural Lifts, a paper on "pair curves".
Paper 3: Smarandache-R-Module and Mcrita Context.
Paper 4: Generalized Vertex Induced Connected Subsets of a Graph.
Paper 5: b-Chromatic Number of Splitting Graph of Wheel.
Paper 6: Eccentric Connectivity and Connective Eccentric Indices.
Paper 7: The Moving Coordinate System and Euler-Savary’s Formula.
Paper 8: Laplacian Energy of Binary Labeled Graph.
Paper 9; Smarandachely total mean cordial labeling, Total Mean Cordial Labeling of Graphs.
Paper 10: Number of Regions in Simple Connected Graph.
Paper 11: Directed Paths.

**Category:** Combinatorics and Graph Theory

[74] **viXra:1505.0175 [pdf]**
*submitted on 2015-05-24 23:28:45*

**Authors:** M. Romig

**Comments:** 4 Pages.

Erdős-Szekeres Theorem is proven. The proof is very similar to the original given by
Erdős and Szekeres. However, it explicitly uses properties of binary trees to prove and
visualize the existence of a monotonic subsequence. It is hoped that this presentation
is helpful for pedagogical purposes.

**Category:** Combinatorics and Graph Theory

[73] **viXra:1505.0167 [pdf]**
*submitted on 2015-05-23 10:29:38*

**Authors:** A. A. Frempong

**Comments:** 14 Pages. Copyright © A. A. Frempong. Paper has been included in vixra:1408.0204 (P vs NP: Solutions of NP Problems by the author))

For one more time, yes, P is equal to NP. For the first time in history, 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 formerly NP-hard problem is now NP-easy problem.
The general approach to solving the different types of NP problems are the same, except that sometimes, specific techniques may differ from each other according to the process involved in the problem. The first step is to arrange the data in the problem in increasing or decreasing order. In the salesman problem, the order will be 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 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.
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 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

[72] **viXra:1504.0232 [pdf]**
*submitted on 2015-04-29 05:17:22*

**Authors:** Carol Lynne Jessop, Paul August Winter

**Comments:** 23 Pages.

The separate study of the two concepts of energy and vertex coverings of graphs has opened many avenues of research. In this paper we combine these two concepts in a ratio, called the eigen-3-cover ratio, to investigate the domination effect of the subgraph induced by a vertex 3-covering of a graph (called the 3-cover graph of ), on the original energy of , where large number of vertices are involved. This is referred to as the eigen-3-cover domination and has relevance, in terms of conservation of energy, when a molecule’s atoms and bonds are mapped onto a graph with vertices and edges, respectively. If this energy-3-cover ratio is a function of , the order of graphs belonging to a class of graph, then we discuss its horizontal asymptotic behavior and attach the graphs average degree to the Riemann integral of this ratio, thus associating eigen-3-cover area with classes of graphs. We found that the eigen-3-cover domination had a strongest effect on the complete graph, while this eigen-3-cover domination had zero effect on star graphs. We show that the eigen-3-cover asymptote of discussed classes of graphs belong to the interval [0,1], and conjecture that the class of complete graphs has the largest eigen-3-cover area of all classes of graphs.

**Category:** Combinatorics and Graph Theory

[71] **viXra:1504.0216 [pdf]**
*submitted on 2015-04-28 00:05:36*

**Authors:** Fu Yuhua

**Comments:** 5 Pages.

With the help of Neutrosophy and Quad-stage Method, the proof for negation of “the four color theorem” is given. In which the key issue is to consider the color of the boundary, thus “the two color theorem” and “the five color theorem” are derived to replace "the four color theorem".

**Category:** Combinatorics and Graph Theory

[70] **viXra:1503.0228 [pdf]**
*submitted on 2015-03-28 15:32:12*

**Authors:** J Gregory Moxness

**Comments:** 321 Pages.

For each of the 480 unique octonion Fano plane mnemonic multiplication tables, there are 7 split octonions (one for each of 7 triads in the parent octonion). This PDF is a comprehensive list of all 3840=480+3360 (octonions + split octonions), their Fano planes, and multiplication tables. They are organized in pairs of 240 parent octonions=(8-bit sign mask)*(30 canonical sets of 7 triads). The pairs of parent octonions are created by flipping (reversing) the first triad (center circular line) creating a unique Fano plane mnemonic.

**Category:** Combinatorics and Graph Theory

[69] **viXra:1503.0148 [pdf]**
*submitted on 2015-03-19 03:21:55*

**Authors:** Paul August Winter, Carol Lynne Jessop

**Comments:** 21 Pages.

The separate study of the two concepts of energy and vertex coverings of graphs has opened many avenues of research. In this paper we combine these two concepts in a ratio, called the eigen-cover ratio, to investigate the domination effect of the subgraph induced by a vertex covering of a graph (called the cover graph of ), on the original energy of , where large number of vertices are involved. This is referred to as the eigen-cover domination and has relevance, in terms of conservation of energy, when a molecule’s atoms and bonds are mapped onto a graph with vertices and edges, respectively. If this energy-cover ratio is a function of , the order of graphs belonging to a class of graph, then we discuss its horizontal asymptotic behavior and attach the graphs average degree to the Riemann integral of this ratio, thus associating eigen-cover area with classes of graphs. We found that the eigen-cover domination had a strongest effect on the complete graph, while this chromatic-cover domination had zero effect on star graphs. We show that the eigen-cover asymptote of discussed classes of graphs belong to the interval [0,1], and conjecture that the class of complete graphs has the largest eigen-cover area of all classes of graphs.

**Category:** Combinatorics and Graph Theory

[68] **viXra:1503.0124 [pdf]**
*submitted on 2015-03-16 04:59:31*

**Authors:** Paul August Winter

**Comments:** 18 Pages.

The graph theoretical ratio, the tree-cover ratio, involving spanning trees of a graph G, and a 2-vertex covering (a minimum set S of vertices such that every edge (or path on 2 vertices) of G has at least vertex end in S) of G has been researched. In this paper we introduce a ratio, called the tree-3-covering ratio with respect to S, involving spanning trees and a 3-vertex covering (a minimum set S of vertices of G such that every path on 3 vertices has at least one vertex in S) of graphs. We discuss the asymptotic convergence of this tree-3-cover ratio for classes of graphs, which may have application in ideal communication situations involving spanning trees and 3-vertex coverings of extreme networks. We show that this asymptote lies on the interval with the dumbbell graph (a complete graph on n-1 vertices appended to an end vertex) has tree-3-cover asymptotic convergence of 1/e, identical to the convergence in the secretary problem, and the tree-cover asymptotic convergence of complete graphs. We also introduce the idea of a tree-3-cover area by integrating this tree-3-cover ratio.
AMS classification: 05C99
Key words: spanning trees of graphs, vertex cover, 3-vertex cover, ratios, social interaction, network communication, convergence, asymptotes.

**Category:** Combinatorics and Graph Theory

[67] **viXra:1503.0046 [pdf]**
*submitted on 2015-03-07 06:42:57*

**Authors:** Paul August Winter

**Comments:** 23 Pages.

Much research has been done involving the chromatic number of a graph involving the least number of colors, that the vertices of a graph can be colored, so that no two adjacent vertices have the same color. The idea of how the chromatic number of a vertex cover of a graph dominates the vertex cover of the original graph, where a large number of vertices are involved, has been investigated. The difference between the energy of the complete graph,, and the energy of any other graph G. has been studied, in terms of a ratio. The complete graph, on n vertices, has chromatic number n, and is significant in terms of its easily accessible graph theoretical properties, such as its high level of connectivity and robustness. In this paper, we introduce a ratio, the chromatic-complete difference ratio, involving the difference between the chromatic number of the complete graph, and the chromatic number of any other connected graph G, on the same number n of vertices. This allowed for the investigation of the effect of the chromatic number of G, with respect to the complete graph, when a large number of vertices are involved - referred to as the chromatic-complete difference domination effect. The value of this domination effect lies on the interval [0,1], with most classes of graphs taking on the right hand end-point, while graphs with a large clique takes on the left hand end-point. When this ratio is a function f(n), of the order of a graph, we attach the average degree of G to the Riemann integral to investigate the chromatic-complete difference area aspect of classes of graphs. We applied these chromatic-complete difference aspects to complements of classes of graphs.
AMS Classification: 05C50
1Corresponding author: Paul August Winter: Department of Mathematics, Howard College, University of KwaZulu-Natal, Glenwood, Durban, 4041, South Africa;
ORCID ID: 0000-0003-3539;
email: winterp@ukzn.ac.za
Key words: Chromatic number, domination, ratios, domination, asymptotes, areas

**Category:** Combinatorics and Graph Theory

[66] **viXra:1502.0145 [pdf]**
*submitted on 2015-02-17 06:44:25*

**Authors:** Paul August Winter, Samson Ojako Dickson

**Comments:** 22 Pages.

The energy of a graph is related to the sum of -electron energy in a molecule represented by a molecular graph and originated by the HMO (Hückel molecular orbital) theory. Advances to this theory have taken place which includes the difference of the energy of graphs and the energy formation difference between and graph and its decomposable parts. Although the complete graph does not have the highest energy of all graphs, it is significant in terms of its easily accessible graph theoretical properties and has a high level of connectivity and robustness, for example. In this paper we introduce a ratio, the eigen-complete difference ratio, involving the difference in energy between the complete graph and any other connected graph G, which allows for the investigation of the effect of energy of G with respect to the complete graph when a large number of vertices are involved. This is referred to as the eigen-complete difference domination effect. This domination effect is greatest negatively (positively), for a strongly regular graph (star graphs with rays of length one), respectively, and zero for the lollipop graph. When this ratio is a function f(n), of the order of a graph, we attach the average degree of G to the Riemann integral to investigate the eigen-complete difference area aspect of classes of graphs. We applied these eigen-complete aspects to complements of classes of graphs.

**Category:** Combinatorics and Graph Theory

[65] **viXra:1501.0106 [pdf]**
*submitted on 2015-01-09 03:51:07*

**Authors:** A. Antipin

**Comments:** 4 Pages. This article in English. Версия на РУССКОМ ЯЗЫКЕ: http://vixra.org/abs/1412.0181

The obtained combinatorial formulas describing random walks on a simple cubic grid.
For the case of 2 dimensions - accurate and simple.
For the case of 3 dimensions - accurate, but, unfortunately, not compact.

**Category:** Combinatorics and Graph Theory

[64] **viXra:1412.0181 [pdf]**
*submitted on 2014-12-15 07:01:38*

**Authors:** A. Antipin

**Comments:** 4 Pages. In the Russian language. The English translation will follow.

The obtained combinatorial formulas describing random walks on a simple cubic grid.
For the case of 2 dimensions - accurate and simple.
For the case of 3 dimensions - accurate, but, unfortunately, not compact.

**Category:** Combinatorics and Graph Theory

[63] **viXra:1411.0562 [pdf]**
*submitted on 2014-11-26 05:08:12*

**Authors:** Paul August Winter, Carol Lynne Jessop, Costas Zachariades

**Comments:** 32 Pages.

Much research has involved the consideration of graphs which have sub-graphs of a particular kind, such as cliques. Known classes of graphs which are eigen-bi-balanced, i.e. they have a pair a,b of non-zero distinct eigenvalues, whose sum and product are integral, have been investigated. In this paper we will define ta new class of graphs, called q-cliqued graphs, on vertices, which contain q cliques each of order q connected to a central vertex, and then prove that these q-cliqued graphs are eigen-bi-balanced with respect to a conjugate pair whose sum is -1 and product 1-q. These graphs can be regarded as design graphs, and we use a specific example in an entomological experiment.
AMS Classification: 05C50
Key words: cliques, eigen-bi-balanced graphs, conjugate pair, designs.

**Category:** Combinatorics and Graph Theory

[62] **viXra:1411.0315 [pdf]**
*submitted on 2014-11-19 05:12:27*

**Authors:** Paul August Winter, Carol Lynne Jessop, Fadekemi Janet Adewusi

**Comments:** 20 Pages.

The complete graph is often used to verify certain graph theoretical definitions and applications. Regarding the adjacency matrix, associated with the complete graph, as a circulant matrix, we find its eigenvalues, and use this result to generate a trigonometrical unit-equations involving the sum of terms of the form , where a is odd. This gives rise to t-complete-eigen sequences and diagrams, similar to the famous Farey sequence and diagram. We show that the ratio, involving sum of the terms of the t-complete eigen sequence, converges to ½ , and use this ratio to find the t-complete eigen area. To find the eigenvalues, associated with the characteristic polynomial of complete graph, using induction, we create a general determinant equation involving the minor of the matrix associated with this characteristic polynomial.

**Category:** Combinatorics and Graph Theory

[61] **viXra:1411.0191 [pdf]**
*submitted on 2014-11-15 14:38:12*

**Authors:** Linfan Mao

**Comments:** 125 Pages. Edited By The Madis of Chinese Academy of Sciences and Beijing University of Civil Engineering and Architecture.

The International J.Mathematical Combinatorics (ISSN 1937-1055) is a fully refereed@@ international journal, sponsored by the MADIS of Chinese Academy of Sciences and published in USA quarterly comprising 100-150 pages approx. per volume, which
publishes original research papers and survey articles in all aspects of Smarandachemulti-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences.

**Category:** Combinatorics and Graph Theory

[60] **viXra:1411.0190 [pdf]**
*submitted on 2014-11-15 14:40:18*

**Authors:** Linfan Mao

**Comments:** 129 Pages. Edited By The Madis of Chinese Academy of Sciences and Beijing University of Civil Engineering and Architecture.

The International J.Mathematical@@ Combinatorics (ISSN 1937-1055) is a fully refereed international journal, sponsored by the MADIS of Chinese Academy of Sciences and published in USA quarterly comprising 100-150 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandachemulti-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences.

**Category:** Combinatorics and Graph Theory

[59] **viXra:1411.0189 [pdf]**
*submitted on 2014-11-15 14:41:29*

**Authors:** Linfan Mao

**Comments:** 111 Pages. Edited By The Madis of Chinese Academy of Sciences and Beijing University of Civil Engineering and Architecture.

The International J.Mathematical Combinatorics (ISSN 1937-1055) is a fully refereed international journal, sponsored by the MADIS of Chinese Academy of Sciences and published in USA quarterly comprising 100-150 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandachemulti-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences.

**Category:** Combinatorics and Graph Theory

[58] **viXra:1411.0050 [pdf]**
*submitted on 2014-11-06 18:55:14*

**Authors:** Ihsan Raja Muda Nasution

**Comments:** 1 Page.

In this paper, we propose an idea of TSP-algorithm for any graph.

**Category:** Combinatorics and Graph Theory

[57] **viXra:1409.0165 [pdf]**
*submitted on 2014-09-24 01:48:20*

**Authors:** Ton Kloks, Yue-Li Wang

**Comments:** 172 Pages. This is the text of a course on various techniques applied in algorithmic graph theory.

This is a course on advances in graph algorithms that we taught in Taiwan. The topics included are Exact algorithms, Graph classes, fixed-parameter algorithms, and graph decompositions.

**Category:** Combinatorics and Graph Theory

[56] **viXra:1409.0162 [pdf]**
*submitted on 2014-09-23 09:26:39*

**Authors:** Bambore Dawit

**Comments:** 9 Pages. we need to follow instruction in each step for detail calculations

This paper shows the non-existence of magic square of squares in order
three by investigating two new tools, the first is representing three perfect
squares in arithmetic progression by two numbers and the second is realizing
the impossibility of two similar equations for the same problem at the same
time in different ways and the variables of one is relatively less than the
other.

**Category:** Combinatorics and Graph Theory

[55] **viXra:1409.0113 [pdf]**
*submitted on 2014-09-14 08:55:20*

**Authors:** Paul August Winter

**Comments:** 21 Pages.

The study of the chromatic number and vertex coverings of graphs has opened many avenues of research. In this paper we combine these two concepts in a ratio, to investigate the domination effect of the chromatic number, of the subgaph induced by a vertex covering of a graph G (called the cover graph of G), on the original chromatic number of G, where large number of vertices are involved. This is referred to as the chromatic-cover domination. If this chromatic-cover ratio is a function of n, the order of graphs belonging to a class of graph, then we discuss its horizontal asymptotic behavior and attach the graphs average degree to the Riemann integral of this ratio, thus associating chromatic-cover area with classes of graphs. We found that the chromatic-cover domination had a strongest effect on complete graph, while this chromatic-cover domination had zero effect on star graphs. We show that the chromatic-cover asymptote of all classes of graphs belong to the interval [0,1], and conjecture that complete graphs are the only class of graphs having chromatic-cover asymptote of 1 and that they also have the largest area . We construct a class of graphs, using known classes of graph where vertices are replaced with cliques on q vertices, thus generating sequences which converges to the chromatic-cover asymptote of known classes of graphs. We use a particular sequence to construct a Farey chromatic-cover sequence which is a subsequence of the famous Farey sequence.

**Category:** Combinatorics and Graph Theory

[54] **viXra:1408.0204 [pdf]**
*submitted on 2014-08-28 23:13:13*

**Authors:** A. A. Frempong

**Comments:** 18 Pages. Copyright © A. A. Frempong

Best news. After over 30 years of debating, the debate is over. Yes, P is equal to NP.
For the first time, NP problems have been solved in this paper. Techniques and formulas were developed and used to solve these problems as well as produce simple equations to help programmers apply the techniques. The techniques and formulas are based on an extended Ashanti fairness wisdom as exemplified below. If two people A and B are to divide items of different sizes which are arranged from the largest size to the smallest size, the procedure will be as follows. In the first round, A chooses the largest size, followed by B choosing the next largest size. In the second round, B chooses first, followed by A. In the third round, A chooses first, followed by B and the process continues up to the last item. To abbreviate the sequence in the above choices, one obtains the sequence "A, BB, AA, BB, AA". Let A and B divide the sum of the whole numbers, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1 as equally as possible, by merely always choosing the largest number. Then A chooses 10, B chooses 9 and 8, followed by A choosing 7 and 6; followed by B choosing 5 and 4; followed by A choosing 3 an 2; and finally, B chooses 1. The sum of A's choices is 10 +7+ 6 + 3 + 2 = 28; and the sum of B's choices is 9 + 8 + 5 + 4 + 1 = 27, with error, plus or minus 0.5 . Observe the sequence "A, BB, AA, BB, AA". Observe also that the sequence is not "AB, AB, AB, AB, AB as one might think. If one were to use AB, AB, AB, AB, AB, the sum for A would be 10 + 8 + 6 + 4 + 2 = 30 and the sum for B would be 9 +7 + 5 +3 + 1 = 25, with error, plus or minus 2.5. The reason why the sequence is "A, BB, AA, BB, AA", and not "AB, AB, AB, AB, AB" is as follows. In the first round, when A chooses first, followed by B, A has the advantage of choosing the larger number and B has the disadvantage of choosing the smaller number. In the second round, if A were to choose first, A would have had two consecutive advantages, and therefore, in the second round, B will choose first to produce the sequence AB, B or ABB. In the third round, A chooses first, because B chose first in the second round. After three rounds, the sequence would be A, BB, AA. When this technique was applied to 100 items of different values or masses, by mere combinations, the total value or mass of A's items was equal to the total value or mass of B's items. Similar results were obtained for 1000 items.
By hand, the techniques can be used to prepare final exam schedules for 100 or 1000 courses. Confirmed is the notion that a method that solves one of these problems can also solve other NP problems. Since six 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

[53] **viXra:1408.0040 [pdf]**
*submitted on 2014-08-07 22:36:44*

**Authors:** Oh Jung Uk

**Comments:** 8 Pages.

The point of inside closed curve could not directly connect to the point of outside. If a point of outside closed curve directly connect to points on closed curve then the maximum number of color of map is 4 under only 3 conditions, and the end point of closed curve is located inside newly created closed curve. If another outside point is added then we could prevent maximum color from exceeding 4 by using to rearrange colors of existing points.

**Category:** Combinatorics and Graph Theory

[52] **viXra:1404.0447 [pdf]**
*submitted on 2014-04-23 11:36:30*

**Authors:** John Frederick Sweeney

**Comments:** 18 Pages.

Pascal’s Triangle, originally Mount Meru of Vedic Physics, provides the perfect format for a combinatorial Universe, with its binomial coefficients, as well as its ease of determining Fibonacci Numbers. Matrix and Clifford algebras, in the form of the chart above, can be shaped into a form identical with Pascal’s Triangle. At the same time, a Romanian researcher has devised an algorithm for determining a Fibonacci Number as a quarternion. This paper poses the question as to whether the Clifford Pyramid contains properties similar to Pascal's Triangle.

**Category:** Combinatorics and Graph Theory

[51] **viXra:1404.0066 [pdf]**
*submitted on 2014-04-08 12:32:15*

**Authors:** Dhananjay P. Mehendale

**Comments:** 9 pages.

In this paper we discuss new approach to deal with k-clique problems or their equivalents, namely, k-independent set problems.

**Category:** Combinatorics and Graph Theory

[50] **viXra:1403.0965 [pdf]**
*submitted on 2014-03-28 22:18:01*

**Authors:** Dhananjay P. Mehendale

**Comments:** 2 pages,

Reconstruction conjecture asks whether it is possible to reconstruct a unique (up to isomorphism) graph from set of its one vertex deleted subgraphs. We show here the validity of reconstruction conjecture for every connected graph which is uniquely reconstructible from the set of all its spanning trees. We make use of a well known result, namely, the reconstruction of a tree from the deck of its pendant point deleted subtrees.

**Category:** Combinatorics and Graph Theory

[49] **viXra:1401.0130 [pdf]**
*submitted on 2014-01-17 19:44:31*

**Authors:** Zhang Tianshu

**Comments:** 21 Pages.

A contact border of two adjacent figures can only be two adjacent borderlines. Let us consider the plane of any uncolored planar map as which consists of two kinds’ parallel straight linear segments according to a strip of a kind alternating a strip of another, and every straight linear segment of each kind consists of two kinds of colored points according to a colored point of a kind alternating a colored point of another, either kind of colored points at a straight linear segment is not alike to either kind of colored points at either adjacent straight linear segment of the straight linear segment. Anyhow the plane has altogether four kinds of colored points.
At the outset, we need transform and classify figures at an uncolored planar map. First merge orderly each figure which adjoins at most three figures and an adjacent figure which adjoins at least four figures into a figure. Secondly merge each tract of figures which adjoin at most three figures and an adjacent figure into a figure. After that, transform every borderline closed curve of figures which compose directly the merging figure into the frame of a rectangle which has only longitudinal and transversal sides, according to the sequence from outside merging figure to inside merging figure. Finally color each figure with a color according to either a color of some particular points of a rectangular borderlines closed curve of the figure, or a color unlike colors of its adjacent figures.

**Category:** Combinatorics and Graph Theory

[42] **viXra:1604.0111 [pdf]**
*replaced on 2016-05-07 19:06:06*

**Authors:** A. A. Frempong

**Comments:** 8 Pages. Copyright © by A. A. Frempong. Reference: P vs NP:Solutions of NP Problems,viXra:1408.0204 by A. A. Frempong

This paper covers the principles and procedures for producing the solution of a problem given the procedure for checking the solution of the problem and vice versa. If a problem can be checked in polynomial time, it can be solved in polynomial time, provided a complete checking procedure is available. From a point A, if one uses one's feet to measure a certain distance by counting steps forwards to a point B, and one wants to check the correctness of the measurement, one would count backwards from the point B using one's feet to see if one returns to exactly the point A. If one returns to A, the forward counting is correct, otherwise it is incorrect. If one counted backwards first from the point B to the point A, one could also count forwards from A to B. Before computers were used in filing taxes in the United States, when one prepared a tax return and wanted to check for arithmetic errors, one would reverse the arithmetic steps from the last arithmetic statement backwards all the way to the first entry on the tax form; and if one obtains a zero after reversing the steps, one was sure that there were no arithmetic errors on the tax form (That is, one began with zero entry going forward and one returned with a zero entry). So also, if one is able to check quickly and completely, the correctness of the solution to a problem, one should also be able to produce the solution of the problem by reversing the steps of the checking process while using opposite operations in each step. If a complete checking process is available, the solution process can be obtained by reversing the steps of the checking while using opposite operations in each step. In checking the correctness of the solution to a problem, one should produce the complete checking process which includes the end of the problem and the beginning of the problem. Checking only the final answer or statement is incomplete checking. Since the solution process and the checking process are inverses of each other, knowing one of them, one can obtain the other by reversing the steps while using opposite operations. To facilitate complete checking, the question should always be posed such that one is compelled to show a checking procedure from which the solution procedure can be deduced. Therefore P is always equal to NP.

**Category:** Combinatorics and Graph Theory

[41] **viXra:1604.0111 [pdf]**
*replaced on 2016-05-06 03:21:13*

**Authors:** A. A. Frempong

**Comments:** 8 Pages. Copyright © by A. A. Frempong. Reference: P vs NP:Solutions of NP Problems,viXra:1408.0204 by A. A. Frempong

This paper covers the principles and procedures for producing the solution of a problem given the procedure for checking the solution of the problem and vice versa. If a problem can be checked in polynomial time, it can be solved in polynomial time, provided a complete checking procedure is available. From a point A, if one uses one's feet to measure a certain distance by counting steps forwards to a point B, and one wants to check the correctness of the measurement, one would count backwards from the point B using one's feet to see if one returns to exactly the point A. If one returns to A, the forward counting is correct, otherwise it is incorrect. If one counted backwards first from the point B to the point A, one could also count forwards from A to B. Before computers were used in filing taxes in the United States, when one prepared a tax return and wanted to check for arithmetic errors, one would reverse the arithmetic steps from the last arithmetic statement backwards all the way to the first entry on the tax form; and if one obtains a zero after reversing the steps, one was sure that there were no arithmetic errors on the tax form (That is, one began with zero entry going forward and one returned with a zero entry). So also, if one is able to check quickly and completely, the correctness of the solution to a problem, one should also be able to produce the solution of the problem by reversing the steps of the checking process while using opposite operations in each step. If a complete checking process is available, the solution process can be obtained by reversing the steps of the checking while using opposite operations in each step. In checking the correctness of the solution to a problem, one should produce the complete checking process which includes the end of the problem and the beginning of the problem. Checking only the final answer or statement is incomplete checking. Since the solution process and the checking process are inverses of each other, knowing one of them, one can obtain the other by reversing the steps while using opposite operations. To facilitate complete checking, the question should always be posed such that one is compelled to show a checking procedure from which the solution procedure can be deduced. Therefore P is always equal to NP.

**Category:** Combinatorics and Graph Theory

[40] **viXra:1604.0111 [pdf]**
*replaced on 2016-04-11 03:15:58*

**Authors:** A. A. Frempong

**Comments:** 8 Pages. Copyright © by A. A. Frempong. Reference: P vs NP:Solutions of NP Problems,viXra:1408.0204 by A. A. Frempong

This paper covers the principles and procedures for producing the solution of a problem given the procedure for checking the solution of the problem and vice versa. If a problem can be checked in polynomial time, it can be solved in polynomial time, provided a complete checking procedure is available. From a point A, if one uses one's feet to measure a certain distance by counting steps forwards to a point B, and one wants to check the correctness of the measurement, one would count backwards from the point B using one's feet to see if one returns to exactly the point A. If one returns to A, the forward counting is correct, otherwise it is incorrect. If one counted backwards first from the point B to the point A, one could also count forwards from A to B. Before computers were used in filing taxes in the United States, when one prepared a tax return and wanted to check for arithmetic errors, one would reverse the arithmetic steps from the last arithmetic statement backwards all the way to the first entry on the tax form; and if one obtains a zero after reversing the steps, one was sure that there were no arithmetic errors on the tax form (That is, one began with zero entry going forward and one returned with a zero entry). So also, if one is able to check quickly the correctness of the solution to a problem, one should also be able to produce the solution of the problem by reversing the steps of the checking process while using opposite operations in each step. If a complete checking process is available, the solution process can be obtained by reversing the steps of the checking while using opposite operations in each step. In checking the correctness of the solution to a problem, one should produce the complete checking process which includes the end of the problem and the beginning of the problem. Checking only the final answer or statement is incomplete checking. Since the solution process and the checking process are inverses of each other, knowing one of them, one can obtain the other by reversing the steps while using opposite operations. To facilitate complete checking, the question should always be posed such that one is compelled to show a checking procedure from which the solution procedure can be deduced. Therefore P is always equal to NP.

**Category:** Combinatorics and Graph Theory

[39] **viXra:1601.0200 [pdf]**
*replaced on 2016-01-30 07:39:24*

**Authors:** Prashanth R. Rao

**Comments:** 3 Pages.

In a previous paper we described a method to represent graph information as a single numerical value by distinctly labeling each of its vertices with unique primes. In this paper, we modify the previous approach to again represent a graph as a single numeric value, we log transform this value and approximate it with an optimum value which if minimized by appropriate prime labeling of the graph should allow us to compare it with another graph on which an identical algorithm is implemented. Identical optimum value minima is a necessary but not sufficient condition for graph isomorphism.

**Category:** Combinatorics and Graph Theory

[38] **viXra:1601.0200 [pdf]**
*replaced on 2016-01-20 19:35:39*

**Authors:** Prashanth R. Rao

**Comments:** 3 Pages.

In a previous paper we described a method to represent graph information as a single numerical value by distinctly labeling each of its vertices with unique primes. In this paper, we modify the previous approach to again represent a graph as a single numeric value, we log transform this value and approximate it with an optimum value which if minimized by appropriate prime labeling of the graph should allow us to compare it with another graph on which an identical algorithm is implemented. Identical optimum value minima may be expected to indicate graph isomorphism.

**Category:** Combinatorics and Graph Theory

[37] **viXra:1512.0322 [pdf]**
*replaced on 2015-12-21 02:59:29*

**Authors:** Dhananjay P. Mehendale

**Comments:** 8 Pages. Typos corrected. Added two more examples.

In this paper we develop a novel characterization for isomorphism of graphs. The characterization is obtained in terms of ordered adjacency lists to be associated with two given labeled graphs. We show that the two given labeled graphs are isomorphic if and only if their associated ordered adjacency lists can be made identical by applying the action of suitable transpositions on any one of these lists. We discuss in brief the complexity of the algorithm for deciding isomorphism of graphs and show that it is of the order of the cube of number of the number of edges.

**Category:** Combinatorics and Graph Theory

[36] **viXra:1512.0222 [pdf]**
*replaced on 2015-12-09 16:10:09*

**Authors:** Prashanth R. Rao

**Comments:** 2 Pages.

We present a simple theoretical strategy to represent using a single numerical value “A” called the prime vertex labeling Adjacency value product, all structural information encoded in a graph. This strategy has the potential to allow us to reconstruct the graph in its entirety based on a single number. To do so we assume that we have access to a large list of prime numbers which are infinite in number. This method will allow us to store graph backbone as a numerical value for retrieval and re-use and may also allow development of algorithms that exploit this representation feature as shortcut to address graph isomorphism.

**Category:** Combinatorics and Graph Theory

[35] **viXra:1512.0222 [pdf]**
*replaced on 2015-12-07 13:16:41*

**Authors:** Prashanth R. Rao

**Comments:** 2 Pages.

We present a simple theoretical strategy to represent using a single numerical value “A” called the prime vertex labeling Adjacency value product, all structural information encoded in a graph. This strategy has the potential to allow us to reconstruct the graph in its entirety based on a single number. To do so we assume that we have access to a large list of prime numbers which are infinite in number. This method will allow us to store graph backbone as a numerical value for retrieval and re-use and may also allow development of algorithms that exploit this representation feature as shortcut to address graph isomorphism.

**Category:** Combinatorics and Graph Theory

[34] **viXra:1511.0225 [pdf]**
*replaced on 2015-12-05 16:57:21*

**Authors:** Richard J. Mathar

**Comments:** 27 Pages. Added Section 6 (cut sets) and removed the inconclusive Appendix B.

A terrace form assigns an integer altitude to each point of a finite two-dimensional square grid
such that the maximum altitude difference between a point and its four neighbors is one.
It is 2-way monotonic if the sign of this altitude difference is zero or one for steps to the East or steps
to the South. We provide tables for the number of 2-way monotonic terrace forms as a function of grid size
and maximum altitude difference, and point at the equivalence to the number of 3-colorings of the grid.

**Category:** Combinatorics and Graph Theory

[33] **viXra:1508.0201 [pdf]**
*replaced on 2015-08-30 18:35:17*

**Authors:** Marco Ripà

**Comments:** 9 Pages.

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 n3 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 hu(n) − hl(n) 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

[32] **viXra:1508.0085 [pdf]**
*replaced on 2015-08-27 14:39:38*

**Authors:** Philip Gibbs

**Comments:** 16 Pages.

The Ulam numbers form an increasing sequence beginning 1,2 such that each subsequent number can be uniquely represented as the sum of two smaller Ulam numbers. An algorithm is described and implemented in Java to compute the first billion Ulam numbers.

**Category:** Combinatorics and Graph Theory

[31] **viXra:1508.0045 [pdf]**
*replaced on 2015-08-08 10:45:52*

**Authors:** Philip Gibbs

**Comments:** 4 Pages.

A conjecture on the quasi-periodic behaviour of Ulam sequences

**Category:** Combinatorics and Graph Theory

[30] **viXra:1505.0167 [pdf]**
*replaced on 2015-05-28 23:37:40*

**Authors:** A. A. Frempong

**Comments:** 14 Pages. Copyright © A. A. Frempong. Paper has been included in vixra:1408.0204 (Example 7 of P vs NP: Solutions of NP Problems by the author)

For one more time, yes, P is equal to NP. For the first time in history, 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 formerly NP-hard problem is now NP-easy problem.
The general approach to solving the different types of NP problems are the same, except that sometimes, specific techniques may differ from each other according to the process involved in the problem. The first step is to arrange the data in the problem in increasing or decreasing order. In the salesman problem, the order will be 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 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. 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

[29] **viXra:1505.0167 [pdf]**
*replaced on 2015-05-28 14:08:10*

**Authors:** A. A. Frempong

**Comments:** 14 Pages. Copyright © A. A. Frempong. Paper has been included in vixra:1408.0204 (Example 7 of P vs NP: Solutions of NP Problems by the author)

For one more time, yes, P is equal to NP. For the first time in history, 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 formerly NP-hard problem is now NP-easy problem.
The general approach to solving the different types of NP problems are the same, except that sometimes, specific techniques may differ from each other according to the process involved in the problem. The first step is to arrange the data in the problem in increasing or decreasing order. In the salesman problem, the order will be 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 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 may not be unique. 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

[28] **viXra:1505.0167 [pdf]**
*replaced on 2015-05-24 14:33:38*

**Authors:** A. A. Frempong

**Comments:** 14 Pages. Copyright © A. A. Frempong. Paper has been included in vixra:1408.0204 (Example 7 of P vs NP: Solutions of NP Problems by the author)

For one more time, yes, P is equal to NP. For the first time in history, 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 formerly NP-hard problem is now NP-easy problem.
The general approach to solving the different types of NP problems are the same, except that sometimes, specific techniques may differ from each other according to the process involved in the problem. The first step is to arrange the data in the problem in increasing or decreasing order. In the salesman problem, the order will be 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 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.
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

[27] **viXra:1505.0167 [pdf]**
*replaced on 2015-05-23 19:51:37*

**Authors:** A. A. Frempong

**Comments:** 14 Pages. Copyright © A. A. Frempong. Paper has been included in vixra:1408.0204 (Example 7 of P vs NP: Solutions of NP Problems by the author)

For one more time, yes, P is equal to NP. For the first time in history, 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 formerly NP-hard problem is now NP-easy problem.
The general approach to solving the different types of NP problems are the same, except that sometimes, specific techniques may differ from each other according to the process involved in the problem. The first step is to arrange the data in the problem in increasing or decreasing order. In the salesman problem, the order will be 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 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.
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

[26] **viXra:1501.0106 [pdf]**
*replaced on 2015-01-09 08:39:50*

**Authors:** A. Antipin

**Comments:** 4 Pages. This article in English. Версия на РУССКОМ ЯЗЫКЕ: http://vixra.org/abs/1412.0181

The obtained combinatorial formulas describing random walks on a simple cubic grid.
For the case of 2 dimensions - accurate and simple.
For the case of 3 dimensions - accurate, but, unfortunately, not compact.

**Category:** Combinatorics and Graph Theory

[25] **viXra:1412.0181 [pdf]**
*replaced on 2015-01-09 08:42:42*

**Authors:** A. Antipin

**Comments:** 4 Pages. Это статья на РУССКОМ ЯЗЫКЕ. The English version of the article: http://vixra.org/abs/1501.0106

**Category:** Combinatorics and Graph Theory

[24] **viXra:1411.0050 [pdf]**
*replaced on 2015-05-31 03:30:25*

**Authors:** Ihsan Raja Muda Nasution

**Comments:** 2 Pages.

In this paper, we propose an idea of TSP-algorithm for any graph.

**Category:** Combinatorics and Graph Theory

[23] **viXra:1411.0050 [pdf]**
*replaced on 2014-11-08 17:34:50*

**Authors:** Ihsan Raja Muda Nasution

**Comments:** 2 Pages.

In this paper, we propose an idea of TSP-algorithm for any graph.

**Category:** Combinatorics and Graph Theory

[22] **viXra:1408.0204 [pdf]**
*replaced on 2015-05-23 11:39:08*

**Authors:** A. A. Frempong

**Comments:** 33 Pages. Copyright © A. A. Frempong

Best news. After over 30 years of debating, the debate is over. Yes, P is equal to NP.
For the first time, NP problems. including the classic traveling salesman problem have been solved in this paper. The general approach to solving the different types of NP problems are the same, except that sometimes, specific techniques may differ from each other according to the process involved in the problem. Another type of NP problems covered is the division of items of different sizes, masses, or values into equal parts. The techniques and formulas developed for dividing these items into equal parts are based on an extended Ashanti fairness wisdom as exemplified below. If two people A and B are to divide items of different sizes which are arranged from the largest size to the smallest size, the procedure would be as follows. In the first round, A chooses the largest size, followed by B choosing the next largest size. In the second round, B chooses first, followed by A. In the third round, A chooses first, followed by B and the process continues up to the last item. To abbreviate the sequence in the above choices, one obtains the sequence "AB, BA AB". Let A and B divide the sum of the whole numbers, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1 as equally as possible, by merely always choosing the largest number. Then A chooses 10, B chooses 9 and 8, followed by A choosing 7 and 6; followed by B choosing 5 and 4; followed by A choosing 3 an 2; and finally, B chooses 1. The sum of A's choices is 10 +7+ 6 + 3 + 2 = 28; and the sum of B's choices is 9 + 8 + 5 + 4 + 1 = 27, with error, plus or minus 0.5 . Observe the sequence "AB, BA, AB, BA, AB". Observe also that the sequence is not "AB, AB, AB, AB, AB as one might think. Te reason why the sequence is "AB, BA AB, BA, AB" is as follows. In the first round, when A chooses first, followed by B, A has the advantage of choosing the larger number and B has the disadvantage of choosing the smaller number. In the second round, if A were to choose first, A would have had two consecutive advantages, and therefore, in the second round, B will choose first to produce the sequence AB, BA. In the third round, A chooses first, because B chose first in the second round. After three rounds, the sequence would be AB, BA, AB. When his technique was applied to 100 items of different values or masses, by mere combinations, the total value or mass of A's items was equal to the total value or mass of B's items. Similar results were obtained for 1000 items. By hand, the techniques can be used to prepare final exam schedules for 100 or 1000 courses.
A new approach to solving the traveling salesman problem was used to determine the shortest route to visit nine cities and return to the starting city. The technique covered eliminates a shortcoming of the nearest neighbor approach as well as that of the grouping of the cities. The distances involved were arranged in increasing order and by inspection, ten distances were selected from a set of the shortest 14 distances, intead of the overall set of 45 distances involved. The selected distances were used to construct the shortest route.
Confirmed is the notion that an approach that solves one of these problems can also solve other NP problems. Since six problems from three different areas 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

[21] **viXra:1408.0204 [pdf]**
*replaced on 2015-05-23 02:41:45*

**Authors:** A. A. Frempong

**Comments:** 33 Pages. Copyright © A. A. Frempong

Best news. After over 30 years of debating, the debate is over. Yes, P is equal to NP.
For the first time, NP problems. including the classic traveling salesman problem have been solved in this paper. The general approach to solving the different types of NP problems are the same, except that sometimes, specific techniques may differ from each other according to the process involved in the problem. Another type of NP problems covered is the division of items of different sizes, masses, or values into equal parts. The techniques and formulas developed for dividing these items into equal parts are based on an extended Ashanti fairness wisdom as exemplified below. If two people A and B are to divide items of different sizes which are arranged from the largest size to the smallest size, the procedure would be as follows. In the first round, A chooses the largest size, followed by B choosing the next largest size. In the second round, B chooses first, followed by A. In the third round, A chooses first, followed by B and the process continues up to the last item. To abbreviate the sequence in the above choices, one obtains the sequence "AB, BA AB". Let A and B divide the sum of the whole numbers, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1 as equally as possible, by merely always choosing the largest number. Then A chooses 10, B chooses 9 and 8, followed by A choosing 7 and 6; followed by B choosing 5 and 4; followed by A choosing 3 an 2; and finally, B chooses 1. The sum of A's choices is 10 +7+ 6 + 3 + 2 = 28; and the sum of B's choices is 9 + 8 + 5 + 4 + 1 = 27, with error, plus or minus 0.5 . Observe the sequence "AB, BA, AB, BA, AB". Observe also that the sequence is not "AB, AB, AB, AB, AB as one might think. Te reason why the sequence is "AB, BA AB, BA, AB" is as follows. In the first round, when A chooses first, followed by B, A has the advantage of choosing the larger number and B has the disadvantage of choosing the smaller number. In the second round, if A were to choose first, A would have had two consecutive advantages, and therefore, in the second round, B will choose first to produce the sequence AB, BA. In the third round, A chooses first, because B chose first in the second round. After three rounds, the sequence would be AB, BA, AB. When his technique was applied to 100 items of different values or masses, by mere combinations, the total value or mass of A's items was equal to the total value or mass of B's items. Similar results were obtained for 1000 items. By hand, the techniques can be used to prepare final exam schedules for 100 or 1000 courses.
A new approach to solving the traveling salesman was used to determine the shortest route to visit nine cities and return to the starting city. The technique covered eliminates a shortcoming of the nearest neighbor approach as well as that of the grouping of the cities. The distances involved were arranged in increasing order and by inspection, ten distances were selected from a set of the shortest 14 distances, intead of the overall set of 45 distances involved. The selected distances were used to construct the shortest route.
Confirmed is the notion that an approach that solves one of these problems can also solve other NP problems. Since six problems from three different areas 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

[20] **viXra:1408.0204 [pdf]**
*replaced on 2014-12-14 17:17:00*

**Authors:** A. A. Frempong

**Comments:** 19 Pages. Copyright © A. A. Frempong

Best news. After over 30 years of debating, the debate is over. Yes, P is equal to NP.
For the first time, NP problems have been solved in this paper. Techniques and formulas were developed and used to solve these problems as well as produce simple equations to help programmers apply the techniques. The techniques and formulas are based on an extended Ashanti fairness wisdom as exemplified below. If two people A and B are to divide items of different sizes which are arranged from the largest size to the smallest size, the procedure will be as follows. In the first round, A chooses the largest size, followed by B choosing the next largest size. In the second round, B chooses first, followed by A. In the third round, A chooses first, followed by B and the process continues up to the last item. To abbreviate the sequence in the above choices, one obtains the sequence "AB, BA AB". Let A and B divide the sum of the whole numbers, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1 as equally as possible, by merely always choosing the largest number. Then A chooses 10, B chooses 9 and 8, followed by A choosing 7 and 6; followed by B choosing 5 and 4; followed by A choosing 3 an 2; and finally, B chooses 1.
The sum of A's choices is 10 +7+ 6 + 3 + 2 = 28; and the sum of B's choices is 9 + 8 + 5 + 4 + 1 = 27, with error, plus or minus 0.5 . Observe the sequence "AB, BA, AB, BA, AB". Observe also that the sequence is not "AB, AB, AB, AB, AB as one might think. If one were to use AB, AB, AB, AB, AB, the sum for A would be 10 + 8 + 6 + 4 + 2 = 30 and the sum for B would be 9 +7 + 5 +3 + 1 = 25, with error, plus or minus 2.5. The reason why the sequence is "AB, BA AB, BA, AB", and not "AB, AB, AB, AB, AB" is as follows. In the first round, when A chooses first, followed by B, A has the advantage of choosing the larger number and B has the disadvantage of choosing the smaller number. In the second round, if A were to choose first, A would have had two consecutive advantages, and therefore, in the second round, B will choose first to produce the sequence AB, BA. In the third round, A chooses first, because B chose first in the second round. After three rounds, the sequence would be AB, BA, AB. When this technique was applied to 100 items of different values or masses, by mere combinations, the total value or mass of A's items was equal to the total value or mass of B's items. Similar results were obtained for 1000 items.
By hand, the techniques can be used to prepare final exam schedules for 100 or 1000 courses. Confirmed is the notion that a method that solves one of these problems can also solve other NP problems. Since six 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

[19] **viXra:1408.0204 [pdf]**
*replaced on 2014-09-17 00:14:25*

**Authors:** A. A. Frempong

**Comments:** 19 Pages. Copyright © A. A. Frempong

Best news. After over 30 years of debating, the debate is over. Yes, P is equal to NP.
For the first time, NP problems have been solved in this paper. Techniques and formulas were developed and used to solve these problems as well as produce simple equations to help programmers apply the techniques. The techniques and formulas are based on an extended Ashanti fairness wisdom as exemplified below. If two people A and B are to divide items of different sizes which are arranged from the largest size to the smallest size, the procedure will be as follows. In the first round, A chooses the largest size, followed by B choosing the next largest size. In the second round, B chooses first, followed by A. In the third round, A chooses first, followed by B and the process continues up to the last item. To abbreviate the sequence in the above choices, one obtains the sequence "AB, BA AB". Let A and B divide the sum of the whole numbers, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1 as equally as possible, by merely always choosing the largest number. Then A chooses 10, B chooses 9 and 8, followed by A choosing 7 and 6; followed by B choosing 5 and 4; followed by A choosing 3 an 2; and finally, B chooses 1.
The sum of A's choices is 10 +7+ 6 + 3 + 2 = 28; and the sum of B's choices is 9 + 8 + 5 + 4 + 1 = 27, with error, plus or minus 0.5 . Observe the sequence "AB, BA, AB, BA, AB". Observe also that the sequence is not "AB, AB, AB, AB, AB as one might think. If one were to use AB, AB, AB, AB, AB, the sum for A would be 10 + 8 + 6 + 4 + 2 = 30 and the sum for B would be 9 +7 + 5 +3 + 1 = 25, with error, plus or minus 2.5. The reason why the sequence is "AB, BA AB, BA, AB", and not "AB, AB, AB, AB, AB" is as follows. In the first round, when A chooses first, followed by B, A has the advantage of choosing the larger number and B has the disadvantage of choosing the smaller number. In the second round, if A were to choose first, A would have had two consecutive advantages, and therefore, in the second round, B will choose first to produce the sequence AB, BA. In the third round, A chooses first, because B chose first in the second round. After three rounds, the sequence would be AB, BA, AB. When this technique was applied to 100 items of different values or masses, by mere combinations, the total value or mass of A's items was equal to the total value or mass of B's items. Similar results were obtained for 1000 items.
By hand, the techniques can be used to prepare final exam schedules for 100 or 1000 courses. Confirmed is the notion that a method that solves one of these problems can also solve other NP problems. Since six 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

[18] **viXra:1408.0204 [pdf]**
*replaced on 2014-09-13 17:03:27*

**Authors:** A. A. Frempong

**Comments:** 19 Pages. Copyright © A. A. Frempong

Best news. After over 30 years of debating, the debate is over. Yes, P is equal to NP.
For the first time, NP problems have been solved in this paper. Techniques and formulas were developed and used to solve these problems as well as produce simple equations to help programmers apply the techniques. The techniques and formulas are based on an extended Ashanti fairness wisdom as exemplified below. If two people A and B are to divide items of different sizes which are arranged from the largest size to the smallest size, the procedure will be as follows. In the first round, A chooses the largest size, followed by B choosing the next largest size. In the second round, B chooses first, followed by A. In the third round, A chooses first, followed by B and the process continues up to the last item. To abbreviate the sequence in the above choices, one obtains the sequence "A, BB, AA, BB, AA". Let A and B divide the sum of the whole numbers, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1 as equally as possible, by merely always choosing the largest number. Then A chooses 10, B chooses 9 and 8, followed by A choosing 7 and 6; followed by B choosing 5 and 4; followed by A choosing 3 an 2; and finally, B chooses 1. The sum of A's choices is 10 +7+ 6 + 3 + 2 = 28; and the sum of B's choices is 9 + 8 + 5 + 4 + 1 = 27, with error, plus or minus 0.5 . Observe the sequence "A, BB, AA, BB, AA". Observe also that the sequence is not "AB, AB, AB, AB, AB as one might think. If one were to use AB, AB, AB, AB, AB, the sum for A would be 10 + 8 + 6 + 4 + 2 = 30 and the sum for B would be 9 +7 + 5 +3 + 1 = 25, with error, plus or minus 2.5. The reason why the sequence is "A, BB, AA, BB, AA", and not "AB, AB, AB, AB, AB" is as follows. In the first round, when A chooses first, followed by B, A has the advantage of choosing the larger number and B has the disadvantage of choosing the smaller number. In the second round, if A were to choose first, A would have had two consecutive advantages, and therefore, in the second round, B will choose first to produce the sequence AB, B or ABB. In the third round, A chooses first, because B chose first in the second round. After three rounds, the sequence would be A, BB, AA. When this technique was applied to 100 items of different values or masses, by mere combinations, the total value or mass of A's items was equal to the total value or mass of B's items. Similar results were obtained for 1000 items.
By hand, the techniques can be used to prepare final exam schedules for 100 or 1000 courses. Confirmed is the notion that a method that solves one of these problems can also solve other NP problems. Since six 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

[17] **viXra:1408.0204 [pdf]**
*replaced on 2014-09-05 15:45:29*

**Authors:** A. A. Frempong

**Comments:** 18 Pages. Copyright © A. A. Frempong

Best news. After over 30 years of debating, the debate is over. Yes, P is equal to NP.
For the first time, NP problems have been solved in this paper. Techniques and formulas were developed and used to solve these problems as well as produce simple equations to help programmers apply the techniques. The techniques and formulas are based on an extended Ashanti fairness wisdom as exemplified below. If two people A and B are to divide items of different sizes which are arranged from the largest size to the smallest size, the procedure will be as follows. In the first round, A chooses the largest size, followed by B choosing the next largest size. In the second round, B chooses first, followed by A. In the third round, A chooses first, followed by B and the process continues up to the last item. To abbreviate the sequence in the above choices, one obtains the sequence "A, BB, AA, BB, AA". Let A and B divide the sum of the whole numbers, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1 as equally as possible, by merely always choosing the largest number. Then A chooses 10, B chooses 9 and 8, followed by A choosing 7 and 6; followed by B choosing 5 and 4; followed by A choosing 3 an 2; and finally, B chooses 1. The sum of A's choices is 10 +7+ 6 + 3 + 2 = 28; and the sum of B's choices is 9 + 8 + 5 + 4 + 1 = 27, with error, plus or minus 0.5 . Observe the sequence "A, BB, AA, BB, AA". Observe also that the sequence is not "AB, AB, AB, AB, AB as one might think. If one were to use AB, AB, AB, AB, AB, the sum for A would be 10 + 8 + 6 + 4 + 2 = 30 and the sum for B would be 9 +7 + 5 +3 + 1 = 25, with error, plus or minus 2.5. The reason why the sequence is "A, BB, AA, BB, AA", and not "AB, AB, AB, AB, AB" is as follows. In the first round, when A chooses first, followed by B, A has the advantage of choosing the larger number and B has the disadvantage of choosing the smaller number. In the second round, if A were to choose first, A would have had two consecutive advantages, and therefore, in the second round, B will choose first to produce the sequence AB, B or ABB. In the third round, A chooses first, because B chose first in the second round. After three rounds, the sequence would be A, BB, AA. When this technique was applied to 100 items of different values or masses, by mere combinations, the total value or mass of A's items was equal to the total value or mass of B's items. Similar results were obtained for 1000 items.
By hand, the techniques can be used to prepare final exam schedules for 100 or 1000 courses. Confirmed is the notion that a method that solves one of these problems can also solve other NP problems. Since six 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

[16] **viXra:1408.0204 [pdf]**
*replaced on 2014-08-31 12:03:19*

**Authors:** A. A. Frempong

**Comments:** 18 Pages. Copyright © A. A. Frempong

**Category:** Combinatorics and Graph Theory