A c-coloring of Gn,m = [n] × [m] is a mapping of Gn,m into [c] such that no four corners forming a rectangle have the same color. In 2009 a challenge was proposed via the internet to find a 4-coloring of G17,17. This attracted considerable attention from the popular mathematics community. A coloring was produced; however, finding it proved to be difficult. The question arises: is the problem of...

The mathematical formalism of quantum resonance combined with tensor product decomposability of unitary evolutions is mapped onto a class of NP-complete combinatorial problems. It has been demonstrated that nature has polynomial resources for solving NP-complete problems and that may help to develop a new strategy for artificial intelligence, as well as to re-evaluate the role of natural select...

Cook [l] has shown that 3-SAT, the Boolean satisfiability problem restricted to instances with exactly three variables per clause, is NP-complete. This is a tightest possible restriction on the number of variables in a clause because as Even et al. [2] demonstrate, 2-SAT is in P. Horowitz and Sahni [5] point up the importance of finding the strongest possible restrictions under which a problem ...

The genus g(G) of a graph G is the smallest number g such that G can be embedded on the orientable surface of genus g. Given a graph G and a natural number k one may ask: Is g(G) I k? This problem, called the graph genus problem, is one of the remaining basic open problems, listed by Garey and Johnson 121, for which there is neither a polynomially bounded algorithm nor a proof that the problem ...

In this paper we restrict every set splitting problem to the special case in which every set has just three elements. This restricted version is also NP-complete. Then, we introduce a general conversion from any set splitting problem to 3-set splitting. Then we introduce a randomize algorithm, and we use Markov chain model for run time complexity analysis of this algorithm. In the last section ...