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

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

**Comments:** 13 Pages.

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

**Category:** Combinatorics and Graph Theory

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

**Authors:** John Archie Gillis

**Comments:** 30 Pages.

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

**Category:** Combinatorics and Graph Theory