Combinatorics and Graph Theory

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Recent submissions

Any replacements are listed farther down

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

Interval Complex Neutrosophic Graph of Type 1

Authors: Said Broumi, Assia Bakali, Mohamed Talea, Florentin Smarandache, V. Venkateswara Rao
Comments: 20 Pages. Neutrosophic Operational Research Volume III

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

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

Independence of Clique Problems

Authors: Thinh Nguyen
Comments: 27 Pages.

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

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

Linear Programming Solves Biclique Problems, Flaws in Literature Proof

Authors: Thinh D. Nguyen
Comments: 2 Pages.

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

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

Exact Weight Perfect Matching of Bipartite Graph Problem Simplified

Authors: Thinh D. Nguyen
Comments: 2 Pages.

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

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

Proof of the Goldbach Conjecture

Authors: Zhuwenhao
Comments: 3 Pages.

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

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

CSP Solver and Capacitated Vehile Routing Problem

Authors: Thinh D. Nguyen
Comments: 10 Pages.

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

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

Graph Isomorphism and Circuit Isomorphism

Authors: Thinh Duc Nguyen
Comments: 1 Page.

In this note, we show that graph isomorphism is reducible to circuit isomorphism, in polynomial time.
Category: Combinatorics and Graph Theory

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

Envp, Another Prime Number Based Strategy to Encode Graphs

Authors: Prashanth R. Rao
Comments: 2 Pages.

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

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

Labeled Trees with Fixed Node Label Sum

Authors: Richard J. Mathar
Comments: 70 Pages.

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

[136] viXra:1805.0079 [pdf] submitted on 2018-05-02 17:54:35

Introduction and Some Results on the Graph Theory for Interval Valued Neutrosophic Sets

Authors: Said Broumi, Mohamed Talea, Assia Bakali, Florentin Smarandache, Quek Shio Gai, Ganeshsree Selvachandran
Comments: 26 Pages. the paper is an extended version

In this paper, motivated by the notion of generalized single-valued neutrosophicgraphs of the first type, we define a new type of neutrosophic graph called the generalized interval- valued neutrosophic graph of first type (GIVNG1) and presented a matrix representation for this graph. Some of the fundamental properties and characteristics of this new concept is also studied. The concept of GIVNG1 is an extension of generalized fuzzy graphs (GFG1) and generalized single-valued neutrosophic graphs of the first type (GSVNG1).
Category: Combinatorics and Graph Theory

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

Bipolar Neutrosophic Minimum Spanning Tree

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

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

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

Zastosowanie Teorii Grafów do Analizy Stabilności Stanów Stacjonarnych W Sieciach Reakcji Enzymatycznch [praca Doktorska, 1978] ### Application of Graph Theory to Stability Analysis of Steady States in Enzymatic Reaction Networks [PhD Thesis, 1978]

Authors: Zbigniew Osiak
Comments: 156 Pages. In Polish

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

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

On the Division of Planar Graphs in Consecutive Prime Parts

Authors: Juan Moreno Borrallo
Comments: 4 Pages.

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

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

On Status Indices of Some Graphs

Authors: Sudhir Jog, Shrinath Patil
Comments: 9 Pages.

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

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

A Paper on Union Closed Families

Authors: Ero Sukna
Comments: 7 Pages.

Timestamp : 20/Feb/2018
Category: Combinatorics and Graph Theory

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

Cause/effect Correlations Through the Borsuk-Ulam Theorem and Kneser Graphs

Authors: Arturo Tozzi
Comments: 10 Pages.

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

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

An Inquiry on the Implications of Abandoning Vixra and Using Arxiv on Finding the Shortest Path Within a Graph

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

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

[128] viXra:1801.0051 [pdf] submitted on 2018-01-05 21:59:11

A Lemma on the Minimal Counter-example of Frankl's Conjecture

Authors: Ankush Hore
Comments: 5 Pages.

Frankl's Conjecture, from 1979, states that any finite union-closed family, containing at least one non-empty member set, must have an element which belongs to at least half of the member-sets. In this paper we list out some properties of the hypothetical minimal counter-example to this conjecture. In particular, we discuss the frequency of 3 distinct elements in the minimal counter-example. We also apply these findings to finite bipartite graphs.
Category: Combinatorics and Graph Theory

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

The Exact Solution of Gauss’s Problem on the Number of Integer "Points" in a Circular and Spherical "Layers"

Authors: Arsen A. Movsesyan
Comments: 23 Pages.

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

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

A Note on Vertex Transitivity in Isomorphic Graphs

Authors: Murugesan, Anitha
Comments: 9 Pages. The last theorem is the highlight of the paper

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

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

Mathematical Combinatoric Fields

Authors: Paris Samuel Miles-Brenden
Comments: 2 Pages. None.

Note on Combinatorics.
Category: Combinatorics and Graph Theory

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

Simple Solution of Traveling Salesman Problem

Authors: Choe Ryujin
Comments: 6 Pages.

Simple solution of traveling salesman problem
Category: Combinatorics and Graph Theory

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

Generalized Interval Valued Neutrosophic Graphs of First Type

Authors: Said Broumi, Assia Bakali, Mohamed Talea, Ali Hassan, Florentin Smarandache
Comments: 7 Pages. 2017 IEEE International Conference on INnovations in Intelligent SysTems and Applications (INISTA), Gdynia Maritime University, Gdynia, Poland, 3-5 July 2017.

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

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

Computation of Shortest Path Problem in a Network with SV-Triangular Neutrosophic Numbers

Authors: Said Broumi, Assia Bakali, Mohamed Talea, Florentin Smarandache
Comments: 6 Pages. 2017 IEEE International Conference on INnovations in Intelligent SysTems and Applications (INISTA), Gdynia Maritime University, Gdynia, Poland, 3-5 July 2017.

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

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

Complex Neutrosophic Graphs of Type 1

Authors: Said Broumi, Assia Bakali, Mohamed Talea, Florentin Smarandache
Comments: 6 Pages. 2017 IEEE International Conference on INnovations in Intelligent SysTems and Applications (INISTA), Gdynia Maritime University, Gdynia, Poland, 3-5 July 2017

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

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

Statistics on Small Graphs

Authors: Richard J. Mathar
Comments: Pages 47-120 are adjacency matrices for connected simple graphs up to 7 nodes.

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

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

Traveling Salesman Problem Solved with Zero Error

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

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

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

The nxnxn Dots Problem Optimal Solution

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

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

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

New Lower Bounds for Van der Waerden Numbers

Authors: Alexey V. Komkov
Comments: 2 Pages.

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

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

Decision-Making Method based on Neutrosophic Soft Expert Graphs

Authors: Vakkas Uluçay, Mehmet Şahin, Said Broumi, Assia Bakali, Mohamed Talea, Florentin Smarandache
Comments: 30 Pages. this article is submited to an elsevier journal

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

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

New Lower Bounds for Van Der Waerden Numbers Using Genetic Algorithm

Authors: Alexey V. Komkov
Comments: Pages.

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

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

A Neutrosophic Graph Similarity Measures

Authors: Shimaa Fathi, Hewayda Elghawalby, A.a. Salama
Comments: 8 Pages.

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

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

Foundation for Neutrosophic Mathematical Morphology

Authors: Eman.m.el-Nakeeb, H. Elghawalby, A.a.salama, S.a.el-Hafeez
Comments: 17 Pages.

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

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

Spectra of New Join of Two Graphs

Authors: Renny P Varghese, K Rejikumar
Comments: 8 Pages.

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

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

Spectra of a New Join in Duplication Graph

Authors: K. Reji Kumar, Renny P. Varghese
Comments: 10 Pages.

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

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

Spectrum of (K; r) Regular Hypergraph

Authors: K Reji Kumar, Renny P Varghese
Comments: 8 Pages.

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

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

The Premature State of “Topology”and “Graph Theory” Nourished by “Seven Bridges of Königsberg Problem”

Authors: Damodar Rajbhandari
Comments: 9 Pages.

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

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

Tiling Hexagons with Smaller Hexagons and Unit Triangles

Authors: Richard J. Mathar
Comments: 10 pages are Java source code distributed under the LGPL 3.

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

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

An Algorithm for Solving the Graph Isomorphism Problem

Authors: Lucas Allen
Comments: English, 9 pages, matrix equations and examples

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

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

Intuition-Based ai for Solutions of NP-Complete Problems.

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

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

Sequences of Primes Obtained by the Method of Concatenation (Collected Papers)

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

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

International Journal of Mathematical Combinatorics, Vol. 1/2016

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

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

Binding Number of Some Special Classes of Trees

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

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

N∗C∗− Smarandache Curve of Bertrand Curves Pair According to Frenet Frame

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

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

On Net-Regular Signed Graphs

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

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

Quotient Cordial Labeling of Graphs

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

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

Mathematical Combinatorics (International Book Series)

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

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

P vs NP Problem Solutions Generalized

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

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

Geodetic Games

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

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

On Bipolar Single Valued Neutrosophic Graphs

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

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

Exact Minimum Lower Bound Algorithm for Traveling Salesman Problem

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

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

Mathematics for Everything with Combinatorics on Nature. a Report on the Promoter Dr. Linfan Mao of Mathematical Combinatorics

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

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

Interval Valued Neutrosophic Graphs

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

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

On Strong Interval Valued Neutrosophic Graphs

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

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

Single Valued Neutrosophic Graphs

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

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

International Journal of Mathematical Combinatorics, Vol. 4, 2015

Authors: Many Authors
Comments: 145 Pages.

Collection of papers on combinatorics.
Category: Combinatorics and Graph Theory

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

A Proof of the Four Color Theorem by Induction

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

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

Defining a Modified Adjacency Value Product Following Unique Prime Labeling of Graph Vertices and Undertaking a Small Step Toward Possible Application for Testing Graph Isomorphism

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

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

Export a Sequence of Prime Numbers

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

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

Removing Magic from the Normal Distribution and the Stirling and Wallis Formulas.

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

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

Isomorphism of Graphs using Ordered Adjacency List

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

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

A Prime Number Based Strategy to Label Graphs and Represent Its Structure as a Single Numerical Value

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

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

Counting 2-way Monotonic Terrace Forms over Rectangular Landscapes

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

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

A Class of Multinomial Permutations Avoiding Object Clusters

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

Testing 4-Critical Plane and Projective Plane Multiwheels Using Mathematica

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

Ripà’s Conjectures on the K-Dimensions 3 X 3 X … X 3 Dots Problem

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

A Computer Program to Solve Water Jug Pouring Puzzles.

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

The n X n X n Points Problem Optimal Solution

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

An Efficient Method for Computing Ulam Numbers

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

A Conjecture for Ulam Sequences

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

Mathematical Combinatorics, Book Series, Vol. 2, 2015

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

A Note on Erdős-Szekeres Theorem

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

P vs NP: Solutions of the Traveling Salesman Problem

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

The Eigen-3-Cover Ratio of Graphs: Asymptotes, Domination and Areas

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

Negating Four Color Theorem with Neutrosophy and Quad-stage Method

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

The Comprehensive Split Octonions and their Fano Planes

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

The Eigen-Cover Ratio of Graphs: Asymptotes, Domination and Areas

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

Tree-3-Cover Ratio of Graphs: Asymptotes and Areas

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

The Chromatic-Complete Difference Ratio of Classes of Graphs- Domination, Asymptotes and Area

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

The Eigen-Complete Difference Ratio of Classes of Graphs Domination, Asymptotes and Area

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

Exact Solution of the Problem of Random Walks on 2 and 3-Dimensional2015-01-08 Simple Cubic Grids in the Form of Combinatorial Expressions. (En)

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

Exact Solution of the Problem of Random Walks on 2-and 3-Dimensional Simple Cubic Grids in the Form of Combinatorial Expressions.

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

The Class of Q-Cliqued Graphs: Eigen-bi-Balanced Characteristic, Designs and an Entomological Experiment

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

The Complete Graph: Eigenvalues, Trigonometrical Unit-Equations with Associated T-Complete-Eigen Sequences, Ratios, Sums and Diagrams

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

International Journal of Mathematical Combinatorics, 1/2014

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

International Journal of Mathematical Combinatorics, 2/2014

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

International Journal of Mathematical Combinatorics, 3/2014

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

The Minimum Sum

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

Advances in Graph Algorithms

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

The Proof for Non-existence of Magic Square of Squares in Order Three

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

The Chromatic-Covering of a Graph: Ratios, Domination, Areas and Farey Sequences

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

Solutions of NP Problems (P vs NP)

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

Try to re-Prove 4-Color Theorem

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

Fibonacci Quarternions

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

On the K-Clique Problems: a New Approach

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

On the Reconstruction of Graphs

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

A Written Proof of the Four-Colors Map Problem

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

[48] viXra:1311.0117 [pdf] submitted on 2013-11-17 05:07:57

The Innovation Game Theory

Authors: Adefokun Tomiwa Michael, Adefokun Tayo Charles
Comments: 4 Pages.

The Innovation Game Theory (IGT) is designed to help businesses, organizations and communities in formulating an approach to their innovation activities in the form of scientific game. As such, leaders, growth managers, theorists, inventors, innovators and consumers visualize the process of overcoming challenges their businesses or communities face (known or unknown) as though it is a game where all strategic actions and resources involved in tackling the challenges are expressed in terms of quantifiable values which are ultimately tied to certain competitive reward mechanisms.
Category: Combinatorics and Graph Theory

[47] viXra:1311.0044 [pdf] submitted on 2013-11-06 09:31:14

A New Calculus on the Ring of Symmetric Functions and Its Applications

Authors: Yusuke Iwahashi
Comments: 7 Pages.

This paper develops a new calculus on the ring of symmetric functions and introduces its application. In the last of this paper, the author describes a new general method to expand any symmetric function in terms of a basis in the ring of symmetric functions. For application of it, the author also mentions a general way to evaluate the transition matrix between any two bases in the ring of symmetric functions.
Category: Combinatorics and Graph Theory

Replacements of recent Submissions

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

Graph Isomorphism and Circuit Isomorphism

Authors: Thinh D. Nguyen
Comments: 4 Pages.

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

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

Envp, Another Prime Number Based Strategy to Encode Graphs

Authors: Prashanth R. Rao
Comments: 2 Pages.

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

[52] viXra:1805.0079 [pdf] replaced on 2018-05-07 05:34:04

Introduction and Some Results on the Graph Theory for Interval Valued Neutrosophic Sets

Authors: Said Broumi, Mohamed Talea, Assia Bakali, Florentin Smarandach, Quek ShioGai, Ganeshsree Selvachandran
Comments: 24 Pages.

In this paper, motivated by the notion of generalized single-valued neutrosophicgraphs of the first type, we define a new type of neutrosophic graph called the generalized interval- valued neutrosophic graph of first type (GIVNG1) and presented a matrix representation for this graph. Some of the fundamental properties and characteristics of this new concept is also studied. The concept of GIVNG1 is an extension of generalized fuzzy graphs (GFG1) and generalized single-valued neutrosophic graphs of the first type (GSVNG1).
Category: Combinatorics and Graph Theory

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

On the Division of Plane Figures in Consecutive Prime Parts

Authors: Juan Moreno Borrallo
Comments: 4 Pages.

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

[50] viXra:1801.0051 [pdf] replaced on 2018-01-07 11:57:20

A Lemma on the Minimal Counter-example of Frankl's Conjecture

Authors: Ankush Hore
Comments: 5 Pages.

Frankl's Conjecture, from 1979, states that any finite union-closed family, containing at least one non-empty member set, must have an element which belongs to at least half of the member-sets. In this paper we list out some properties of the hypothetical minimal counter-example to this conjecture. In particular, we discuss the frequency of 3 distinct elements in the minimal counter-example. We also apply these findings to finite bipartite graphs.
Category: Combinatorics and Graph Theory

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

The Exact Solution of Gauss’s Problem on the Number of Integer "Points" in a Circular and Spherical "Layers"

Authors: Arsen A. Movsesyan
Comments: 23 Pages.

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

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

Exact Map Inference in General Higher-Order Graphical Models Using Linear Programming

Authors: Ikhlef Bechar
Comments: 50 Pages.

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

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

Traveling Salesman Problem Solved with Zero Error

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

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

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

New Lower Bounds for Van der Waerden Numbers

Authors: Alexey V. Komkov
Comments: 2 Pages.

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

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

Hamiltonian Paths in Graphs

Authors: Atul Mehta
Comments: 7 Pages.

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

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

Constructing a Four-Colorable Map

Authors: Ihsan Raja Muda Nasution
Comments: 2 Pages.

In this paper, we solve the four-color problem by a new algorithm.
Category: Combinatorics and Graph Theory

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

P vs NP Problem Solutions Generalized

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

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

P vs NP Problem Solutions Generalized

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 2016-04-11 03:15:58

P vs NP Problem Solutions Generalized

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

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

Defining a Modified Adjacency Value Product Following Unique Prime Labeling of Graph Vertices and Undertaking a Small Step Toward Possible Application for Testing Graph Isomorphism

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

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

Defining a Modified Adjacency Value Product Following Unique Prime Labeling of Graph Vertices and Undertaking a Small Step Toward Possible Application for Testing Graph Isomorphism

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

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

Isomorphism of Graphs using Ordered Adjacency List

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

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

A Prime Number Based Strategy to Label Graphs and Represent Its Structure as a Single Numerical Value

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

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

A Prime Number Based Strategy to Label Graphs and Represent Its Structure as a Single Numerical Value

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

Counting 2-way Monotonic Terrace Forms over Rectangular Landscapes

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

[34] viXra:1509.0140 [pdf] replaced on 2016-12-01 05:58:16

A Computer Program to Solve Water Jug Pouring Puzzles.

Authors: Richard J. Mathar
Comments: 33 Pages. Added more references and more explicit solutions in Version 2.

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

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

The n X n X n Points Problem Optimal Solution

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

An Efficient Method for Computing Ulam Numbers

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

A Conjecture for Ulam Sequences

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

P vs NP: Solutions of the Traveling Salesman Problem

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

P vs NP: Solutions of the Traveling Salesman Problem

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

P vs NP: Solutions of the Traveling Salesman Problem

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

P vs NP: Solutions of the Traveling Salesman Problem

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

Exact Solution of the Problem of Random Walks on 2 and 3-Dimensional Simple Cubic Grids in the Form of Combinatorial Expressions. (En)

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

Exact Solution of the Problem of Random Walks on 2-and 3-Dimensional Simple Cubic Grids in the Form of Combinatorial Expressions. (Ru)

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

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

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

The Minimum Sum

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

The Minimum Sum

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

P vs NP: Solutions of NP Problems

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

P vs NP: Solutions of NP Problems

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

P vs NP:Solutions of NP Problems

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

P vs NP: Solutions of NP Problems

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

P vs NP: Solutions of NP Problems

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

P vs NP: Solutions of NP Problems

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

P vs NP: Solutions of NP Problems

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