Summary: In memory eta studying various aspects of polimonio of tutte a tessellation regulate began introducing some definitions and rsultados significant Theory Grafos. First we focus on the calculation of polynomial Tutte of teselaciones or masaicos the plane through squares, triangles, hexagons and possible combinations of these three types of polígones regular, in addition to studying various problems listed in several areas of mathematics related this polinomio.Ofrecemos a system that allows encoding structures in principle as diverse as the teselaciones regualeres the plane, and an effective algorithm that automates the call recursive definition of polynomial Tutte, and allowing the calculation of the polynomial fragments teselaciones large dimensions. Emphasize the importance of this algorithm as it allows for the first time reaches dimendiones considerable, not only in fragments of the square mesh, but in fragments through any tessellation flat squares, triangles and hexágonos.Este breakthrough supposed to improve the heights limits aasintóticos and able to obtain significant results in problems enumeration in theory Grafos: calculating orentaciones acíclicas and calculating numbers Whitney, and in two classic problems of geometry, the calculation of the number of cells that arrangements hiperplanos divide spaces euclídeos high dimensions and cáculo the number of vertices of a zonotopo. Leaving aside the issues of enumeration, we wonder to what extent the polynomial Tutte determines the graph to which it is asociado.Esta issue arises due to the amount of invariants associated with a graph which is asociado.Esta question arises because of the many invariants associated with a graph contained in the polynomial Tutte.En this report we demonstrate the existence of large families of graphs that have uniqueness of this property, which fenominamos Tutte unicidad.Las three families studied graphs are locally cuadriculados, teselaciones hexagonal and graphs locally $ C-6 $. All of them have in common ownership lser locally flat and this will allow us to prove that all these families graphs are locally eastern necessary tool to demonstrate their Tutte unicidad.En First we tested the graphs locamente cuadriculados are unÂ'çimvocmente determined by the polynomial Tutte.A then we classify the teselacions hexagonal and graphs locally $ C-6 $. These classifications assume great importance, apart from by its own complexity, it rectification of the classification given by Tomasen in
1991.Estudiamos also invariant associated with these different families of graphs and demonstrated the existence of a relationship with the minor graphs cuadriculados locally. Finally, we set all the machinery necessary to demonstrate the uniqueness of this family Tutte and because of the similarity of the remaining cases, which do not provide anything new from a mathematical point of view, we focus on demonstrating the uniqueness of the tessellation Tutte Hexagonal toroidal.
