In this paper we present a correlation between a variation of the list problem coloring in graphs, the (γ, μ)-coloring, and the property of choosability in graphs, resulting in the k-(γ, μ)-choosability. The list coloring problem is a variation of the classical vertex coloring problem, introduced by Erdos et al. in 1979, along with a property very studied in list coloring: the choosability in graphs. In this work, algorithms were developed to determine the k-choosability and the k-(γ, μ)-choosability of a general simple graph, and a greedy heuristic based on DSATUR. In addition, we applied special techniques to prove the property of choosability in some classes of graphs, involving a general proof for simple graphs, which guarantees that if a graph is k-colorable it is also k-(γ, μ)-choosable, being the computational complexity reduced from the class Πp2-complete to the NP-complete.

In this work we consider a Quasi-Newton optimization algorithm for solving the Molecular Distance Geometry Problem (MDGP). We will deal with the problem through its continuous nature.

The problem of estimating the full three-dimensional structure of a molecule, determining the position in space of all the atoms that compose it, is called Molecular Distance Geometry Problem (MDGP). To do this from an incomplete set of distances is NP-hard computational problem, where to get a feasible solution in a reasonable execution time presenting interesting mathematical and computational challenges. In this work, continuous and discrete mathematical approaches to solve MDGP is revised, based on the analysis of two types of calculating of sphere intersection: solving nonlinear systems from interatomic Euclidean distance equations, or solving internal coordinate systems using matrix multiplication techniques. We adapted the Branch-and-Prune (BP) method considering four spheres intersection. Computational experiments using instances from PDB benchmark are performed, determining the 3D structure based on our theoretical assumptions in a competitive computational processing time.