Three Problems in Graph Coteries to Show P Does not Equal NP

The P versus NP problem, a conjecture formulated by Stephen Cook in 1971, is one of the most challenging problems in contemporary mathematics and theoretical computer science. A concise mathematical formulation of the problem reads: is P = NP? In longer phrasing, this asks: given a problem instance, if some additional data can be recognized fast enough as logically implying the existence of a solution (to the instance), then can a solution be computed fast enough without the aid of any such additional data? In this article we present the idea of graph coteries that are sets whose members are sets with speci ed graph-theoretic properties. Then we formulate three problems in graph coteries to show P does not equal NP.