**Previous months:**

2009 - 0908(1)

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

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

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

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

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

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

2016 - 1601(3) - 1602(6) - 1603(2) - 1604(8) - 1605(2) - 1606(1) - 1607(1) - 1608(1) - 1609(1) - 1610(3) - 1611(3) - 1612(1)

2017 - 1701(1)

Any replacements are listed further down

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

**Authors:** Alexey V. Komkov

**Comments:** 2 Pages.

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

**Category:** Combinatorics and Graph Theory

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

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

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

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** Alexey V. Komkov

**Comments:** Pages.

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

**Category:** Combinatorics and Graph Theory

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

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

**Comments:** 8 Pages.

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

**Category:** Combinatorics and Graph Theory

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

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

**Comments:** 17 Pages.

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

**Category:** Combinatorics and Graph Theory

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

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

**Comments:** 8 Pages.

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

**Category:** Combinatorics and Graph Theory

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

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

**Comments:** 10 Pages.

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

**Category:** Combinatorics and Graph Theory

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

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

**Comments:** 8 Pages.

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** Damodar Rajbhandari

**Comments:** 9 Pages.

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** Richard J. Mathar

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

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** Lucas Allen

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

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** Michail Zak

**Comments:** 15 Pages.

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** Marius Coman

**Comments:** 151 Pages.

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** Linfan Mao

**Comments:** 141 Pages.

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

**Category:** Combinatorics and Graph Theory

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

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

**Comments:** 6 Pages.

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

**Category:** Combinatorics and Graph Theory

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

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

**Comments:** 7 Pages.

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** Nutan G. Nayak

**Comments:** 8 Pages.

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

**Category:** Combinatorics and Graph Theory

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

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

**Comments:** 8 Pages.

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** Linfan MAO

**Comments:** 141 Pages.

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** A. A. Frempong

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

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

**Category:** Combinatorics and Graph Theory

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

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

**Comments:** 12 Pages.

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** Ali Reza Najar Saligheh

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

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

**Category:** Combinatorics and Graph Theory

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

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

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

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** Mohamed Eleiche

**Comments:** 12 Pages.

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** Florentin Smarandache

**Comments:** 4 Pages.

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

**Category:** Combinatorics and Graph Theory

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

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

**Comments:** 33 pages

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

**Category:** Combinatorics and Graph Theory

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

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

**Comments:** 21 pages

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

**Category:** Combinatorics and Graph Theory

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

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

**Comments:** 16 pages

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** Many Authors

**Comments:** 145 Pages.

Collection of papers on combinatorics.

**Category:** Combinatorics and Graph Theory

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

**Authors:** Quang Nguyen Van

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

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** Prashanth R. Rao

**Comments:** 3 Pages.

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** Quang Nguyen Van

**Comments:** 5 Pages.

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** Mikhail Kovalyov

**Comments:** 8 Pages.

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** Dhananjay P. Mehendale

**Comments:** 6 pages

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** Prashanth R. Rao

**Comments:** 2 Pages.

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** Richard J. Mathar

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

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** Ali Reza Najar Saligheh

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

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** Richard J. Mathar

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

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** Dainis Zeps

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

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** Marco Ripà

**Comments:** 5 Pages.

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** Richard J. Mathar

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

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** Marco Ripà

**Comments:** Pages.

We provide an optimal strategy to solve the n X n X n points problem inside the box, considering only 90° and 45° turns, and at the same time a pattern able to drastically lower down the known upper bound. We use a very simple spiral frame, especially if compared to the previous plane by plane approach, that significantly reduces the number of straight lines connected at their end-points necessary to join all the n3 dots, for any n > 5. In the end, we combine the square spiral frame with the rectangular spiral pattern in the most profitable way, in order to minimize the difference h_u(n) − h_l(n) between the upper and the lower bound, proving that it is ≤ 0.5 ∙ n ∙ (n + 3), if n > 1.

**Category:** Combinatorics and Graph Theory

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

**Authors:** Philip Gibbs

**Comments:** 16 Pages.

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** Philip Gibbs

**Comments:** 2 Pages.

A conjecture on the quasi-periodic behaviour of Ulam sequences

**Category:** Combinatorics and Graph Theory

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

**Authors:** editor Linfan Mao

**Comments:** 152 Pages.

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** M. Romig

**Comments:** 4 Pages.

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** A. A. Frempong

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

For one more time, yes, P is equal to NP. For the first time in history, the traveling salesman can determine by hand, with zero or negligible error, the shortest route from home base city to visit once, each of three cities, 10 cities, 20 cities, 100 cities, or 1000 cities, and return to the home base city. The formerly NP-hard problem is now NP-easy problem.
The general approach to solving the different types of NP problems are the same, except that sometimes, specific techniques may differ from each other according to the process involved in the problem. The first step is to arrange the data in the problem in increasing or decreasing order. In the salesman problem, the order will be increasing order, since one's interest is in the shortest distances. The main principle here is that the shortest route is the sum of the shortest distances such that the salesman visits each city once and returns to the starting city. The shortest route to visit nine cities and return to the starting city was found in this paper. It was also found out that even though the length of the shortest route is unique, the sequence of the cities involved is not unique.
Since an approach that solves one of these problems can also solve other NP problems. and the traveling salesman problem has been solved, all NP problems can be solved, provided one has an open mind and continues to think. If all NP problems can be solved, then all NP problems are P problems and therefore, P is equal to NP. The CMI Millennium Prize requirements have been satisfied..

**Category:** Combinatorics and Graph Theory

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

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

**Comments:** 23 Pages.

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** Fu Yuhua

**Comments:** 5 Pages.

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** J Gregory Moxness

**Comments:** 321 Pages.

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

**Category:** Combinatorics and Graph Theory

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

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

**Comments:** 21 Pages.

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** Paul August Winter

**Comments:** 18 Pages.

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** Paul August Winter

**Comments:** 23 Pages.

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

**Category:** Combinatorics and Graph Theory

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

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

**Comments:** 22 Pages.

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** A. Antipin

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

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** A. Antipin

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

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

**Category:** Combinatorics and Graph Theory

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

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

**Comments:** 32 Pages.

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

**Category:** Combinatorics and Graph Theory

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

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

**Comments:** 20 Pages.

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** Linfan Mao

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

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** Linfan Mao

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

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** Linfan Mao

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

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** Ihsan Raja Muda Nasution

**Comments:** 1 Page.

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

**Category:** Combinatorics and Graph Theory

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

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

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

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** Bambore Dawit

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

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** Paul August Winter

**Comments:** 21 Pages.

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** A. A. Frempong

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

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** Oh Jung Uk

**Comments:** 8 Pages.

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** John Frederick Sweeney

**Comments:** 18 Pages.

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** Dhananjay P. Mehendale

**Comments:** 9 pages.

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** Dhananjay P. Mehendale

**Comments:** 2 pages,

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** Zhang Tianshu

**Comments:** 21 Pages.

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

**Category:** Combinatorics and Graph Theory

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

**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*

**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

[46] **viXra:1310.0229 [pdf]**
*submitted on 2013-10-25 13:12:48*

**Authors:** Dhananjay P. Mehendale

**Comments:** 10 pages

We establish the existence of graceful labeling for any unlabeled tree by proposing actual construction procedure for such labeling. We define so called lattice and lattice paths sitting inside it. A lattice path is produced by starting with bottom (or top) row of the lattice and choosing one lattice point per row in the lattice in succession and joining these lattice points. With these lattice points we associate vertex pairs representing edges in a complete graph. It obviously follows that each of so called lattice path represents a graceful graph and further it easily follows that there exist in all n! graceful graphs (among which some are trees) in a complete graph on n vertices. In this paper we propose an algorithm to construct graceful labeling for any given unlabeled tree through construction of appropriate lattice path for this tree under consideration.

**Category:** Combinatorics and Graph Theory

[45] **viXra:1310.0023 [pdf]**
*submitted on 2013-10-05 03:30:14*

**Authors:** Nehul Yadav

**Comments:** 13 Pages. Please reply back to me at nehul12@gmail.com

This research conceptualises the the theories in mathematics with ecology and our evolution. It traces the link between the planet earth with mathematics. It uses calculus , graphs , combinatorics and other theories an models in mathematics. Hope it gets published!

**Category:** Combinatorics and Graph Theory

[44] **viXra:1309.0063 [pdf]**
*submitted on 2013-09-09 11:36:24*

**Authors:** Linfan Mao

**Comments:** 44 Pages.

Study of combinatorial spaces and smarandache multispaces.

**Category:** Combinatorics and Graph Theory

[43] **viXra:1309.0062 [pdf]**
*submitted on 2013-09-09 11:39:24*

**Authors:** Linfan Mao

**Comments:** 56 Pages.

This report survey the smarandache system, labeling graphs and others in the classical graph theory.

**Category:** Combinatorics and Graph Theory

[42] **viXra:1309.0061 [pdf]**
*submitted on 2013-09-09 11:41:53*

**Authors:** Linfan Mao

**Comments:** 56 Pages.

Combinatorial structures underlying things, Topology of Combinatorial Manifold, and Differentiationon Combinatorial Manifold are the topics of this paper.

**Category:** Combinatorics and Graph Theory

[41] **viXra:1309.0060 [pdf]**
*submitted on 2013-09-09 11:46:30*

**Authors:** Linfan Mao

**Comments:** 22 Pages.

An axiom is said smarandachely denied (S-denied)
if in the same space the axiom behaves differently,
i.e., validated and Invalided, or only invalidated
but in at least two distinct ways.
And a Smarandache multi-space is a union of n different spaces equipped with some different structures for an integer n≧2.
This paper studies them.

**Category:** Combinatorics and Graph Theory

[40] **viXra:1309.0059 [pdf]**
*submitted on 2013-09-09 11:48:05*

**Authors:** Linfan Mao

**Comments:** 27 Pages.

I will prescribe limitation on following topics and
mainly concentrate on myself works for combinatorially differential
geometry, particularly, the combinatorial counterpart in recent years.

**Category:** Combinatorics and Graph Theory

[39] **viXra:1309.0058 [pdf]**
*submitted on 2013-09-09 11:49:41*

**Authors:** Linfan Mao

**Comments:** 43 Pages.

Applications of Voltage Assignament
to Principal Fiber Bundles.

**Category:** Combinatorics and Graph Theory

[38] **viXra:1309.0057 [pdf]**
*submitted on 2013-09-09 12:01:40*

**Authors:** Linfan Mao

**Comments:** 36 Pages.

We study the Smarandache Systems with Labeled Topological Graphs.
A rule R in a mathematical system (Sigma; R) is said to be Smarandachely denied if it behaves in at least two different ways within the same set Sigma, i.e. validated and invalided, or only invalided but in multiple distinct ways.
A Smarandache system (Sigma;R) is a system that has at least one smarandachely denied rule R.

**Category:** Combinatorics and Graph Theory

[37] **viXra:1308.0037 [pdf]**
*submitted on 2013-08-06 19:55:15*

**Authors:** Linfan Mao

**Comments:** 73 Pages.

A Smarandache multi-space is a union of n different spaces
equipped with some different structures, for an integer n≧2.

**Category:** Combinatorics and Graph Theory

[36] **viXra:1308.0030 [pdf]**
*submitted on 2013-08-05 21:42:13*

**Authors:** Mohammed Kasim, Fumao Zhang, Qiang Wang

**Comments:** 6 Pages.

Suppose $G$ is a simple graph. The eigenvalues $\delta_1,
\delta_2,\ldots, \delta_n$ of $G$ are the eigenvalues of its
adjacency matrix $A$. The Estrada index of the graph $G$ is defined
as $EE = EE(G) = \Sigma_{i=1}^{n} e^{\delta_i}$. In this paper the
basic properties of $EE$ are investigated. Moreover, some lower and
upper bounds for the Estrada index in terms of the number of
vertices, edges and the Randic index are obtained. In addition, some
relations between $EE$ and graph energy $E(G)$ are presented.

**Category:** Combinatorics and Graph Theory

[35] **viXra:1307.0095 [pdf]**
*submitted on 2013-07-19 21:01:04*

**Authors:** Marco Ripà

**Comments:** The paper is 5 pages long and it is in Italian. Copyright: © 2013 Ripà

Here is the second and last part of the generalized solution to the extended “Nine Dots Puzzle”. In this paper I provide a non-trivial Lower Bound for the k-dimensional n_1 X n_2 X … X n_k points problem. In this way, you can build a range in which certainly will fall all the best possible solutions to the problem we are considering. In conclusion, I provide a few characteristic numerical examples in order to appreciate the quality of the result arising from the particular approach I have chosen.

**Category:** Combinatorics and Graph Theory

[34] **viXra:1307.0055 [pdf]**
*submitted on 2013-07-11 11:56:55*

**Authors:** John Frederick Sweeney

**Comments:** 20 Pages.

In Vedic Physics, the combinatorial process by which counts are counted closely resembles the binomial triangle, or Pascal's Triangle, which was known to Ancient India and China as Mount Meru or Zhang's triangle. This is not accidental, but instead reflects the numerical structure of the universe, as the pattern repeats itself in many types of numbers, including Clifford Algebras, Octonions, the Exceptional Lie Algebras, the Magic Triangle, and Barnes - Wall Lattices. Previous papers have discussed details of the counts; this paper focuses on the triangular relationships between numbers, especially those noted by Tony "Frank" Smith.

**Category:** Combinatorics and Graph Theory

[33] **viXra:1307.0052 [pdf]**
*submitted on 2013-07-10 04:56:05*

**Authors:** John Frederick Sweeney

**Comments:** 11 Pages.

Matter may exist in one of three states in the universe. The interactions between these states of matter add up to fifty types of change. Causes of change include: the spectrum of inter-changed states, due to imbalance, failure to synchronize, balance and coherent synchronization, caused by the interplay of three modes of matter, which sum up to 50 (order of powers). This paper describes the three states of matter and how their interactions combine to create fifty types of change.

**Category:** Combinatorics and Graph Theory

[32] **viXra:1307.0041 [pdf]**
*submitted on 2013-07-08 10:59:13*

**Authors:** John Frederick Sweeney

**Comments:** 9 Pages.

Decoding of Vedic literature reveals three states of matter, and the basic count of matter as modulated by the Golden Section. Thus, all forms of nature follow the contours of the Golden Section in their growth and development. Provided here is a basic explanation of how the counts occur within the three separate states of matter, and a new formation of Time. These concepts form the basis of the Qi Men Dun Jia Model, with its incorporation of the icosahedron and its 60 stellated permutations.

**Category:** Combinatorics and Graph Theory

[31] **viXra:1307.0021 [pdf]**
*submitted on 2013-07-04 10:04:35*

**Authors:** Marco Ripà, Pablo Remirez

**Comments:** The paper is 14 pages long and it is in Italian. Copyright: © 2013 Ripà, Remirez

The classic wit problem, the “Nine Dots Puzzle” is widely used in courses on creativity and appears in a lot of games magazines. One of the earliest appearances is in “Cyclopedia of Puzzles” by Sam Loyd’s in 1914. Here is a review of the generic solution of the problem of the 9 points spread to n2 points. Basing on a specific pattern, we show that any nxn (for n ≥ 5) points puzzle can be also solved Inside the Box, using only 2n−2 straight lines (connected at their end-points), through the square spiral method. The same pattern is also useful to “bound above” the minimal number of straight lines we need to connect nk points in a k-dimensional space, while to “bound below” the solution of the nxnx…xn puzzle we start from a very basic consideration.

**Category:** Combinatorics and Graph Theory

[30] **viXra:1305.0050 [pdf]**
*submitted on 2013-05-08 10:15:41*

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

**Comments:** Pages.

The association of integers, conjugate pairs and tightness with the eigenvalues of graphs provide the motivation for the following definitions. A class of graphs, with the property, that for each graph (member) of the class, there exists a pair a,b of non-zero, distinct, eigenvalues, whose sum (or product) yields the same integer, either as a fixed constant, or a function of an inherent aspect of the graph (such as its size), is said to be sum-eigen*(a+b)*pair balanced (or product-eigen*(a.b)*pair balanced, respectively). For example, complete graphs on n vertices, are eigen-bi-balanced with sum-eigen*(n-2)*pair balanced and product-eigen*(1-n)*pair balanced, and since a,b are non-zero their reciprocals (which affect the tightness of a graph ) are defined, so that this class has the eigen-balanced ratio of 1/a+1/b=(a+b)/(a.b)= (n-2)/(1-n) =f(n) which is asymptotic to the constant value of -1. The absolute value of the integral of f(n) multiplied by the average degree yields the area (n-1)(n-ln(n-1)) – we show that this is the maximum area for most known classes of eigen-bi-balanced graphs. We investigate the effect of this asymptotic ratio on the energy of the molecular representation of graphs. Cycles are generally neither sum-eigen-pair, nor product-eigen-pair balanced, while paths are only sum- eigen-pair balanced. In this paper, we introduce a class of graphs, involving q cliques each of size q, and show that this class is eigen-bi-balanced with respect to the sum -1 and product 1-q so that it has ratio 1/(q-1) asymptotic to 0, and has area q(2q+2ln(q-1)), and discuss its eigen-bi-balanced criticality.

**Category:** Combinatorics and Graph Theory

[29] **viXra:1305.0038 [pdf]**
*submitted on 2013-05-06 20:09:16*

**Authors:** Tom Harvey

**Comments:** 12 Pages.

A self-avoiding walk (SAW) is a path on a lattice that does not pass through the same point more than once. We develop a method for enumerating self-avoiding walks in a lattice by decomposing them into smaller pieces called tiles, solving particular cases on the square, triangular and cubic lattices. We also show that enumeration of SAWs in a lattice is related to enumeration of edge-connected shapes, for example polyominoes.

**Category:** Combinatorics and Graph Theory

[28] **viXra:1304.0004 [pdf]**
*submitted on 2013-04-02 01:26:20*

**Authors:** Okunoye Babatunde O.

**Comments:** 6 Pages.

In computational complexity, the Decision version of the Directed Hamiltonian Path Problem is known to be NP-complete (Nondeterministic-Polynomial complete). There are no known efficient algorithms for its resolution in Polynomial time. In three papers, the author shows that this problem can be resolved in Polynomial time under two special conditions relating to the determinant of a matrix: the absence of zero rows (columns) and similar rows (columns). In this paper, the author gives a brief overview of the proposed solution and the P vs NP problem.

**Category:** Combinatorics and Graph Theory

[27] **viXra:1304.0002 [pdf]**
*submitted on 2013-04-01 07:25:24*

**Authors:** Dhananjay P. Mehendale

**Comments:** 5 pages

The problem of finding shortest Hamiltonian path in a weighted complete graph belongs to the class of NP-Complete problems [1]. In this paper we will show that we can obtain shortest Hamiltonian path in a given weighted complete graph in polynomial time! We will be discussing a very simple but useful idea of applying certain chosen sequence of permutations (actually transpositions) on given weighted adjacency matrix corresponding to the complete graph, on p points say, under consideration. This simple and novel algorithm essentially consists of applying certain transpositions that will transform the weighted adjacency matrix in such a way that its vertices are now relabeled and in this relabeled weighted complete graph the algorithm terminates decisively in producing the shortest Hamiltonian path, and this shortest Hamiltonian path will be
1->2->3->....->(p-1)->p

**Category:** Combinatorics and Graph Theory

[26] **viXra:1303.0172 [pdf]**
*submitted on 2013-03-22 20:54:42*

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

**Comments:** 14 Pages.

Let G and H be two cographs.
We show that the
problem to determine whether H is a retract of G
is NP-complete.
We show that this problem is fixed-parameter tractable when
parameterized by the size of H.
When restricted to the class of threshold graphs or
to the class of
trivially perfect graphs, the problem becomes tractable
in polynomial time. The problem is also
solvable in linear time
when one cograph is given as an induced subgraph
of the other. We characterize absolute retracts
for the class of cographs. Foldings generalize retractions.
We show that the problem to fold a
trivially perfect graph onto a largest possible clique is
NP-complete. For a threshold graph this folding number equals
its chromatic number and achromatic number.

**Category:** Combinatorics and Graph Theory

[25] **viXra:1211.0081 [pdf]**
*submitted on 2012-11-14 10:59:09*

**Authors:** Natasha Lee, Joan Portmann

**Comments:** 5 Pages.

The paper deals with the problems of characterization of simple graphical partitions belonging to the solid graphs, i.e. graphs, in which there are no four of vertices such that it is possible some shift of edges incidental to them and with characterization of the one class of steady graphs too. The necessary and sufficient
conditions for the partition belonging to the solid graph have been established.

**Category:** Combinatorics and Graph Theory

[24] **viXra:1211.0074 [pdf]**
*submitted on 2012-11-13 09:11:35*

**Authors:** Linfan Mao

**Comments:** 46 Pages.

Different from the system in classical mathematics, a
Smarandache system is a contradictory system in which an axiom behaves in at least
two different ways within the same system, i.e., validated and invalided, or
only invalided but in multiple distinct ways. Such systems exist extensively
in the world, particularly, in our daily life. In this paper, we discuss such a
kind of Smarandache system, i.e., non-solvable ordinary differential equation
systems by a combinatorial approach, classify these systems and characterize
their behaviors, particularly, the sum-stability and prod-stability of such
linear and non-linear differential equations. Some applications of such systems
to other sciences, such as those of globally controlling of infectious diseases,
establishing dynamical equations of instable structure, particularly, the
n-body problem and understanding global stability of matters with multilateral
properties can be also found.

**Category:** Combinatorics and Graph Theory

[23] **viXra:1210.0161 [pdf]**
*submitted on 2012-10-27 09:09:57*

**Authors:** Martin Erik Horn

**Comments:** 17 Pages.

The trinomial triangle can be constructed in a binomial way using unit vectors of geometric algebra of quarks. This sheds some light on the question, how it is possible to transform mathematically entities of two elements into entities of three elements or vice versa.

**Category:** Combinatorics and Graph Theory

[22] **viXra:1210.0081 [pdf]**
*submitted on 2012-10-16 23:30:24*

**Authors:** D. B. Chandler, M.-S. Chang, T. Kloks, J. Liu, S.-L. Peng

**Comments:** 121 Pages.

Let GG be a class of graphs. A graph G is a probe graph of GG if its vertex set can be partitioned into a set P of `probes' and an independent set N of `nonprobes' such that G can be embedded into a graph of GG by adding edges between certain nonprobes.
In this book we investigate probe graphs of various classes of graphs.

**Category:** Combinatorics and Graph Theory

[21] **viXra:1210.0049 [pdf]**
*submitted on 2012-10-10 05:51:51*

**Authors:** Okunoye Babatunde O.

**Comments:** 6 Pages.

The decision version of Directed Hamiltonian path problem is an NP-complete problem which asks, given a directed graph G, does G contain a directed Hamiltonian path? In two separate papers, the author expresses the graph problem as an adjacency matrix and a proof given to show that under two special conditions relating to theorems on the determinant of a square matrix, a non-zero determinant value certifies the existence of a directed Hamiltonian path. Here, a brief note is added to repair a flaw in the proof. The result, as expressed in the paper title is a more defensible proposition

**Category:** Combinatorics and Graph Theory

[20] **viXra:1209.0051 [pdf]**
*submitted on 2012-09-17 05:15:21*

**Authors:** Jiawei Gao, Ton Kloks, Sheung-Hung Poon

**Comments:** 15 Pages.

We show that there is a linear-time algorithm to
partition the edges of a planar graph into triangles. We show that the problem is also polynomial for toroidal graphs but NP-complete for
k-planar graphs, where k is at least 8.

**Category:** Combinatorics and Graph Theory

[19] **viXra:1209.0021 [pdf]**
*submitted on 2012-09-06 18:40:36*

**Authors:** Okunoye Babatunde O.

**Comments:** 4 Pages. submitted to IEEE African Journal of Computing and ICTs

In earlier work, the author conjectured that under two special conditions relating to theorems on the determinant of a matrix: the absence of a zero row (column) and the absence of similar rows (columns), a non-zero determinant value certifies the existence of a Directed Hamiltonian Path in an arbitrary adjacency matrix. Here, a formal proof is provided by means of deductive logic to establish that in an arbitrary adjacency matrix of size n (n rows and n columns), a non-zero determinant value verifies the existence of a Directed Hamiltonian Path in the adjacency matrix.

**Category:** Combinatorics and Graph Theory

[18] **viXra:1208.0223 [pdf]**
*submitted on 2012-08-26 22:40:04*

**Authors:** J.W.Nienhuys (Ling-Ju Hung, Ton Kloks eds.)

**Comments:** 192 Pages.

This is a translation of the handwritten classroom notes taken by Nienhuys of a course in combinatorics
given by N.G. de Bruijn at Eindhoven University of
Technology, during the 1970s and 1980s.

**Category:** Combinatorics and Graph Theory

[17] **viXra:1208.0217 [pdf]**
*submitted on 2012-08-24 21:20:18*

**Authors:** Ton Kloks

**Comments:** 18 Pages. In Dutch

In memoriam N.G. de Bruijn. In this paper
I show some highlights of his work in combinatorics. This article does NOT contain an overview of his work in Penrose tilings, asymptotics and AUTOMATH. Other surveys on these
topics are being written by others.

**Category:** Combinatorics and Graph Theory

[16] **viXra:1204.0019 [pdf]**
*submitted on 2012-04-05 03:07:16*

**Authors:** Dhananjay P. Mehendale

**Comments:** 7 pages.

In this paper we obtain a new polynomial time algorithm for testing isomorphism of graphs. This algorithm is based on the idea of associating a rooted, unordered, pseudo tree with given graphs and thus reducing the isomorphism problem for graphs to isomorphism problems for associated rooted, unordered, pseudo trees. We show that isomorphism of the rooted, unordered, pseudo trees associated with graphs and so in effect isomorphism of given two graphs can be tested in polynomial (quadratic) time.

**Category:** Combinatorics and Graph Theory

[15] **viXra:1202.0078 [pdf]**
*submitted on 2012-02-27 01:17:08*

**Authors:** Dainis Zeps

**Comments:** 12 Pages. http://arxiv.org/abs/1202.4862

We build unbounded classes of plane and projective plane multiwheels that are 4-critical that are received summing odd
wheels as edge sums modulo two. These classes can be considered as ascending from single common graph that can be received
as edge sum modulo two of the octahedron graph O and the minimal wheel W3. All graphs of these classes belong to
2n - 2-edges-class of graphs, among which are those that quadrangulate projective plane, i.e., graphs from Groetzsch
class, received applying Mycielski's Construction to odd cycle.

**Category:** Combinatorics and Graph Theory

[14] **viXra:1202.0077 [pdf]**
*submitted on 2012-02-27 01:23:06*

**Authors:** Dainis Zeps

**Comments:** 4 Pages. Full version of paper http://arxiv.org/abs/1202.4862

We build unbounded classes of plane and projective plane multiwheels that are 4-critical that are received summing odd wheels as edge sums modulo two. These classes can be considered as ascending from single common graph that can be received as edge sum modulo two of the octahedron graph O and the minimal wheel W_3.

**Category:** Combinatorics and Graph Theory

[13] **viXra:1106.0053 [pdf]**
*submitted on 27 Jun 2011*

**Authors:** Paul A. Titze

**Comments:** 18 pages, 19 figures.

fix a ship's position in charted interstellar space with the assistance of
a three dimensional computer based stellar chart and star camera spectrometers
capable of measuring angular separations between three sets of
pair stars. The method offers another tool for the navigator to rely on if
alternative position fixing methods are not available or if the navigator
wishes to verify the validity of one's position given by other means.

**Category:** Combinatorics and Graph Theory

[12] **viXra:1103.0032 [pdf]**
*submitted on 11 Mar 2011*

**Authors:** Linfan Mao

**Comments:** 16 pages

An interesting symmetry on multiplication of numbers found by
Prof.Smarandache recently. By considering integers or elements in groups on
graphs, we extend this symmetry on graphs and find geometrical symmetries.
For extending further, Smarandache's or combinatorial systems are also
discussed on general mathematical systems in this paper, particularly, the CC
conjecture presented by myself six years ago, which enables one to construct
symmetrical systems in mathematical sciences.

**Category:** Combinatorics and Graph Theory

[11] **viXra:1101.0095 [pdf]**
*submitted on 28 Jan 2011*

**Authors:** Yilun Shang

**Comments:** 5 pages

An edge-colored graph G is rainbow edge-connected if any two vertices are connected
by a path whose edges have distinct colors. The rainbow connection of a connected
graph G, denoted by rc(G), is the smallest number of colors that are needed in
order to make G rainbow connected. Similarly, a vertex-colored graph G is rainbow
vertex-connected if any two vertices are connected by a path whose internal vertices
have distinct colors. The rainbow vertex-connection of a connected graph G, denoted
by rvc(G), is the smallest number of colors that are needed in order to make G rainbow
vertex-connected. We prove that both rc(G) and rvc(G) have sharp concentration in
classical random graph model G(n, p).

**Category:** Combinatorics and Graph Theory

[10] **viXra:1010.0025 [pdf]**
*submitted on 13 Oct 2010*

**Authors:** Dainis Zeps

**Comments:** 14 pages

We consider combinatorial maps with fixed combinatorial knot numbered with
augmenting numeration called normalized knot. We show that knot's normalization
doesn't affect combinatorial map what concerns its generality. Knot's normalization
leads to more concise numeration of corners in maps, e.g., odd or even corners allow
easy to follow distinguished cycles in map caused by the fixation of the knot.
Knot's normalization may be applied to edge structuring knot too. If both are
normalized then one is fully and other partially normalized mutually.

**Category:** Combinatorics and Graph Theory

[9] **viXra:1009.0014 [pdf]**
*submitted on 13 Mar 2010*

**Authors:** Florentin Smarandache

**Comments:**
3 pages.

Sudoku is a game with numbers, formed by a square with the side of 9, and on each row
and column are placed the digits 1, 2, 3, 4, 5, 6, 7, 8, 9, written only one time; the square is
subdivided in 9 smaller squares with the side of 3x3, which, also, must satisfy the same
condition, i.e. each square to contain all digits from 1 to 9 written only once.

**Category:** Combinatorics and Graph Theory

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

**Authors:** Alexey V. Komkov

**Comments:** 2 Pages.

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** Ihsan Raja Muda Nasution

**Comments:** 2 Pages.

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** A. A. Frempong

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

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** A. A. Frempong

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

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** A. A. Frempong

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

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** Prashanth R. Rao

**Comments:** 3 Pages.

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** Prashanth R. Rao

**Comments:** 3 Pages.

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** Dhananjay P. Mehendale

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

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** Prashanth R. Rao

**Comments:** 2 Pages.

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** Prashanth R. Rao

**Comments:** 2 Pages.

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** Richard J. Mathar

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

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

**Category:** Combinatorics and Graph Theory

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

**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*

**Authors:** Marco Ripà

**Comments:** 9 Pages.

We provide an optimal strategy to solve the n X n X n points problem inside the box, considering only 90° turns, and at the same time a pattern able to drastically lower down the known upper bound. We use a very simple spiral frame, especially if compared to the previous plane by plane approach, that significantly reduces the number of straight lines connected at their end-points necessary to join all the n3 dots. In the end, we combine the square spiral frame with the rectangular spiral pattern in the most profitable way, in order to minimize the difference hu(n) − hl(n) between the upper and the lower bound, proving that it is ≤ 0.5 ∙ n ∙ (n + 3), for any n > 1.

**Category:** Combinatorics and Graph Theory

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

**Authors:** Philip Gibbs

**Comments:** 16 Pages.

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** Philip Gibbs

**Comments:** 4 Pages.

A conjecture on the quasi-periodic behaviour of Ulam sequences

**Category:** Combinatorics and Graph Theory

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

**Authors:** A. A. Frempong

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

For one more time, yes, P is equal to NP. For the first time in history, the traveling salesman can determine by hand, with zero or negligible error, the shortest route from home base city to visit once, each of three cities, 10 cities, 20 cities, 100 cities, or 1000 cities, and return to the home base city. The formerly NP-hard problem is now NP-easy problem.
The general approach to solving the different types of NP problems are the same, except that sometimes, specific techniques may differ from each other according to the process involved in the problem. The first step is to arrange the data in the problem in increasing or decreasing order. In the salesman problem, the order will be increasing order, since one's interest is in the shortest distances. The main principle here is that the shortest route is the sum of the shortest distances such that the salesman visits each city once and returns to the starting city. The shortest route to visit nine cities and return to the starting city was found in this paper. It was also found out that even though the length of the shortest route is unique, the sequence of the cities involved is not unique. Since an approach that solves one of these problems can also solve other NP problems. and the traveling salesman problem has been solved, all NP problems can be solved, provided that one has an open mind and continues to think. If all NP problems can be solved, then all NP problems are P problems and therefore, P is equal to NP. The CMI Millennium Prize requirements have been satisfied..

**Category:** Combinatorics and Graph Theory

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

**Authors:** A. A. Frempong

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

For one more time, yes, P is equal to NP. For the first time in history, the traveling salesman can determine by hand, with zero or negligible error, the shortest route from home base city to visit once, each of three cities, 10 cities, 20 cities, 100 cities, or 1000 cities, and return to the home base city. The formerly NP-hard problem is now NP-easy problem.
The general approach to solving the different types of NP problems are the same, except that sometimes, specific techniques may differ from each other according to the process involved in the problem. The first step is to arrange the data in the problem in increasing or decreasing order. In the salesman problem, the order will be increasing order, since one's interest is in the shortest distances. The main principle here is that the shortest route is the sum of the shortest distances such that the salesman visits each city once and returns to the starting city. The shortest route to visit nine cities and return to the starting city was found in this paper. It was also found out that even though the length of the shortest route is unique, the sequence of the cities involved may not be unique. Since an approach that solves one of these problems can also solve other NP problems. and the traveling salesman problem has been solved, all NP problems can be solved, provided that one has an open mind and continues to think. If all NP problems can be solved, then all NP problems are P problems and therefore, P is equal to NP. The CMI Millennium Prize requirements have been satisfied..

**Category:** Combinatorics and Graph Theory

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

**Authors:** A. A. Frempong

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

For one more time, yes, P is equal to NP. For the first time in history, the traveling salesman can determine by hand, with zero or negligible error, the shortest route from home base city to visit once, each of three cities, 10 cities, 20 cities, 100 cities, or 1000 cities, and return to the home base city. The formerly NP-hard problem is now NP-easy problem.
The general approach to solving the different types of NP problems are the same, except that sometimes, specific techniques may differ from each other according to the process involved in the problem. The first step is to arrange the data in the problem in increasing or decreasing order. In the salesman problem, the order will be increasing order, since one's interest is in the shortest distances. The main principle here is that the shortest route is the sum of the shortest distances such that the salesman visits each city once and returns to the starting city. The shortest route to visit nine cities and return to the starting city was found in this paper. It was also found out that even though the length of the shortest route is unique, the sequence of the cities involved is not unique.
Since an approach that solves one of these problems can also solve other NP problems. and the traveling salesman problem has been solved, all NP problems can be solved, provided that one has an open mind and continues to think. If all NP problems can be solved, then all NP problems are P problems and therefore, P is equal to NP. The CMI Millennium Prize requirements have been satisfied..

**Category:** Combinatorics and Graph Theory

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

**Authors:** A. A. Frempong

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

For one more time, yes, P is equal to NP. For the first time in history, the traveling salesman can determine by hand, with zero or negligible error, the shortest route from home base city to visit once, each of three cities, 10 cities, 20 cities, 100 cities, or 1000 cities, and return to the home base city. The formerly NP-hard problem is now NP-easy problem.
The general approach to solving the different types of NP problems are the same, except that sometimes, specific techniques may differ from each other according to the process involved in the problem. The first step is to arrange the data in the problem in increasing or decreasing order. In the salesman problem, the order will be increasing order, since one's interest is in the shortest distances. The main principle here is that the shortest route is the sum of the shortest distances such that the salesman visits each city once and returns to the starting city. The shortest route to visit nine cities and return to the starting city was found in this paper. It was also found out that even though the length of the shortest route is unique, the sequence of the cities involved is not unique.
Since an approach that solves one of these problems can also solve other NP problems. and the traveling salesman problem has been solved, all NP problems can be solved, provided that one has an open mind and continues to think. If all NP problems can be solved, then all NP problems are P problems and therefore, P is equal to NP. The CMI Millennium Prize requirements have been satisfied..

**Category:** Combinatorics and Graph Theory

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

**Authors:** A. Antipin

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

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** A. Antipin

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** Ihsan Raja Muda Nasution

**Comments:** 2 Pages.

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** Ihsan Raja Muda Nasution

**Comments:** 2 Pages.

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** A. A. Frempong

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

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** A. A. Frempong

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

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** A. A. Frempong

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

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** A. A. Frempong

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

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** A. A. Frempong

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

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** A. A. Frempong

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

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** A. A. Frempong

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

**Category:** Combinatorics and Graph Theory

[15] **viXra:1310.0229 [pdf]**
*replaced on 2013-11-12 01:52:44*

**Authors:** Dhananjay P. Mehendale

**Comments:** 15 Pages. Rivised

We define so called n-delta lattice containing (n-1) lattice points in first (topmost) row, (n-2) lattice points in second row, and so on. Each time the count of lattice points decreases by unity as we move down by one row till we reach the last (bottommost) row containing single lattice point. We label these lattice points in two different ways and obtain two different labeled lattices. In the first kind of labeling we associate vertex pairs in a particular way as labels for points of the lattice and so call it edge-labeled n-delta lattice. In the second kind of labeling we associate integers as labels with lattice points in each row to indicate the position of that lattice point in the row and so call it position-labeled n-delta lattice. This defining of position-labeled n-delta lattice enables us to associate a lexicographic ordering with lattice paths. We define distinct as well as different lattice paths and further see that for proving graceful tree conjecture one needs to show that the count of distinct lattice paths corresponding to trees in the edge-labeled n-delta lattice is same as the count of nonisomorphic trees with n vertices. We verify this for some (small) values of n. We further see that existence of graceful labeling for an unlabeled tree with n vertices follows from the existence of a lattice path representing this same tree in the edge-labeled n-delta lattice. It is possible to generate all (n, n-1)-trees from all (n-1, n-2)-trees by attaching an edge that emerges from each of the inequivalent vertices of (n-1, n-2)-trees and entering in the new vertex taken outside. We show that extending all lattice paths by adding a lattice point in the paths sitting in the sub-lattice of n-delta lattice in all possible ways is same as the above mentioned generation of trees from lower trees.

**Category:** Combinatorics and Graph Theory

[14] **viXra:1307.0095 [pdf]**
*replaced on 2014-01-11 12:49:08*

**Authors:** Marco Ripà

**Comments:** 13 Pages.

A generalization of Ripà’s square spiral solution for the nXnX…X n points upper bound problem. Additionally, we provide a non-trivial lower bound for the k-dimensional n_1Xn_2X…Xn_k points problem. In this way, we can build a range in which, with certainty, all the best possible solutions to the problem we are considering will fall. Finally, we provide a few characteristic numerical examples in order to appreciate the fineness of the result arising from the particular approach we have chosen.

**Category:** Combinatorics and Graph Theory

[13] **viXra:1307.0095 [pdf]**
*replaced on 2013-09-06 17:31:43*

**Authors:** Marco Ripà

**Comments:** 13 Pages.

A generalization of Ripà’s square spiral solution for the nXnX…X n points upper bound problem. Additionally, we provide a non-trivial lower bound for the k-dimensional n_1Xn_2X…Xn_k points problem. In this way, we can build a range in which, with certainty, all the best possible solutions to the problem we are considering will fall. Finally, we provide a few characteristic numerical examples in order to appreciate the fineness of the result arising from the particular approach we have chosen.

**Category:** Combinatorics and Graph Theory

[12] **viXra:1307.0021 [pdf]**
*replaced on 2013-09-05 19:08:16*

**Authors:** Marco Ripà, Pablo Remirez

**Comments:** 11 Pages. This is the second version of the paper in Italian that I already submitted a few months ago. It has been entirely written in English.

The classic thinking problem, the “Nine Dots Puzzle”, is widely used in courses on creativity and appears in a lot of games magazines. One of the earliest appearances is in “Cyclopedia of Puzzles” by Sam Loyd in 1914. Here is a review of the generic solution of the problem of the 9 points spread to n^2 points. Basing it on a specific pattern, we show that any nxn (for n ≥ 5) points puzzle can also be solved ‘Inside the Box’, using only 2∙n − 2 straight lines (connected at their end-points), through the square spiral method. The same pattern is also useful to “bound above” the minimal number of straight lines we need to connect n^k points in a k-dimensional space, while to “bound below” the solution of the nxnx…xn puzzle we start from a very basic consideration.

**Category:** Combinatorics and Graph Theory

[11] **viXra:1307.0021 [pdf]**
*replaced on 2013-07-30 15:01:36*

**Authors:** Marco Ripà, Pablo Remirez

**Comments:** 15 Pages.

The classic wit problem, the “Nine Dots Puzzle” is widely used in courses on creativity and appears in a lot of games magazines. One of the earliest appearances is in “Cyclopedia of Puzzles” by Sam Loyd’s in 1914. Here is a review of the generic solution of the problem of the 9 points spread to n2 points. Basing on a specific pattern, we show that any nxn (for n ≥ 5) points puzzle can be also solved Inside the Box, using only 2n−2 straight lines (connected at their end-points), through the square spiral method. The same pattern is also useful to “bound above” the minimal number of straight lines we need to connect nk points in a k-dimensional space, while to “bound below” the solution of the nxnx…xn puzzle we start from a very basic consideration.

**Category:** Combinatorics and Graph Theory

[10] **viXra:1307.0021 [pdf]**
*replaced on 2013-07-08 18:48:41*

**Authors:** Marco Ripà, Pablo Remirez

**Comments:** The paper is 15 pages long and it is in Italian. Copyright: © 2013 Ripà, Remirez

The classic wit problem, the “Nine Dots Puzzle” is widely used in courses on creativity and appears in a lot of games magazines. One of the earliest appearances is in “Cyclopedia of Puzzles” by Sam Loyd’s in 1914. Here is a review of the generic solution of the problem of the 9 points spread to n2 points. Basing on a specific pattern, we show that any nxn (for n ≥ 5) points puzzle can be also solved Inside the Box, using only 2n−2 straight lines (connected at their end-points), through the square spiral method. The same pattern is also useful to “bound above” the minimal number of straight lines we need to connect nk points in a k-dimensional space, while to “bound below” the solution of the nxnx…xn puzzle we start from a very basic consideration.

**Category:** Combinatorics and Graph Theory

[9] **viXra:1304.0002 [pdf]**
*replaced on 2013-04-11 15:24:12*

**Authors:** Dhananjay P. Mehendale

**Comments:** 14 Pages

The problem of finding shortest Hamiltonian path and shortest Hamiltonian circuit in a weighted complete graph belongs to the class of NP-Complete problems [1]. This well known problem asks for a method or algorithm to locate such path or circuit that passes through every vertex only once in the given weighted complete graph. In this paper we begin with proposing two approximation algorithms for shortest Hamiltonian graphs which essentially consists of applying certain chosen permutations (transpositions or product of transpositions) on the adjacency matrix of given weighted complete graph causing reshuffling of the labels of its vertices. We change the labels of vertices through proper choice of permutations in such a way that in this relabeled graph the Hamiltonian path 123….k(k+1)…p becomes approximation to shortest path in the given weighted complete graph under consideration. We then define so called ordered weighted adjacency list for given weighted complete graph and proceed to the main result of the paper, namely, the exact algorithm based on utilization of ordered weighted adjacency list and the simple properties that any path or circuit must satisfy. This algorithm performs checking of sub-lists, containing (n-1) entries (edge pairs) for paths and n entries (edge pairs) for circuits, chosen from ordered adjacency list in a well defined sequence to determine exactly the shortest Hamiltonian path and shortest Hamiltonian circuit. The procedure has intrinsic advantage of landing on the desired solution in quickest possible time and even in worst case in polynomial time.

**Category:** Combinatorics and Graph Theory

[8] **viXra:1209.0021 [pdf]**
*replaced on 2012-09-11 22:39:16*

**Authors:** Okunoye Babatunde

**Comments:** 4 Pages. Accepted and Revised at IEEE African Journal of Computing and ICTs

In earlier work, the author conjectured that under two special conditions relating to theorems on the determinant of a matrix: the absence of a zero row (column) and the absence of similar rows (columns), a non-zero determinant value certifies the existence of a Directed Hamiltonian Path in an arbitrary adjacency matrix. Here, a formal proof is provided by means of deductive logic to establish that in an arbitrary adjacency matrix of size n (n rows and n columns), a non-zero determinant value verifies the existence of a Directed Hamiltonian Path in the adjacency matrix

**Category:** Combinatorics and Graph Theory

[7] **viXra:1208.0217 [pdf]**
*replaced on 2013-01-06 01:50:48*

**Authors:** Ton Kloks

**Comments:** 18 Pages. in Dutch

In memoriam N.G. de Bruijn. In this article I present some highlights of De Bruijn's contributions in combinatorics. This article does not survey his work on eg Penrose tilings, asymptotics or AUTOMATH; other surveys on these topics are being written by others.

**Category:** Combinatorics and Graph Theory

[6] **viXra:1208.0217 [pdf]**
*replaced on 2013-01-06 00:55:45*

**Authors:** Ton Kloks

**Comments:** 18 Pages. in Dutch

In memoriam N.G. de Bruijn. In this article I present some highlights of De Bruijn's contributions in combinatorics. This article does not survey his work on eg Penrose tilings, asymptotics or AUTOMATH; other surveys on these topics are being written by others.

**Category:** Combinatorics and Graph Theory

[5] **viXra:1208.0217 [pdf]**
*replaced on 2012-12-31 20:30:05*

**Authors:** Ton Kloks

**Comments:** 18 Pages. in Dutch

In memoriam N.G. de Bruijn. In this article I present an short survey of the work of N.G. de Bruijn in combinatorics. This text does not survey his work on asymptotics, Penrose tilings, or AUTOMATH. Surveys covering these topics will appear elsewhere.

**Category:** Combinatorics and Graph Theory

[4] **viXra:1208.0217 [pdf]**
*replaced on 2012-09-30 02:20:29*

**Authors:** T. Kloks

**Comments:** 18 Pages. in Dutch

In memoriam N.G. de Bruijn. In this article I present a short survey of the work of N.G. de Bruijn in
combinatorics. This text does not survey his work
in asymptotics, Penrose tilings, or AUTOMATH. Surveys covering these topics will appear elsewhere.

**Category:** Combinatorics and Graph Theory

[3] **viXra:1208.0217 [pdf]**
*replaced on 2012-09-07 22:43:37*

**Authors:** T. Kloks

**Comments:** 18 Pages. In Dutch

In memoriam: N.G. de Bruijn.
In this short survey I present an overview of
De Bruijn's work in combinatorics.
This text does not survey his work in asymptotics,
Penrose tilings, or AUTOMATH. Surveys covering these topics will appear elsewhere.

**Category:** Combinatorics and Graph Theory