Research Article
Delgado JJ
Abstract
Definition of theorem: On a politic map, the neighbour countries can not to take the same colour because they could seem the same country. When the frontier between countries is a point, we must not consider it as a frontier. This is possible with three and five colours, it is proved, but with four at this moment lack an evident proof, without pc. There are infinite maps where the solution must to run, also inside of them it must to solve the relation between infinite and only four colours. For that I transform politic maps into maps in the plane graph, then I use their polygons to create other new polygons, with a particular centre. Then the group of new polygons form a structure, which distribute all points in two independent substructures, the Centres and the Crowns. This particular centre is the common vertex of some polygons belonging to plane graph, and they form a new polygon. The points which surround the centre constitute a barrier that impede the direct relation between the centre and other points. I call crown to this barrier. Each polygon is linked with the previous, so it achieves shape of spiral. Also it achieves independence among centres, and the points of the same crown do not entwine, their relations are consecutives, two by two. To specify the new structure group points in two substructures, the Centres and the Crowns, one colour goes to the centres, and three to the crowns. On the crown there is a process of colours run on a finite number of points chosen by triangulations, which impose a Stopping Condition. The triangulation happens when two or more points with two different colours have a common neighbour, then this point must to take the third colour. On each process after last triangulation happen always stopping condition, it mean that neighbour points without colours have two possibilities, and the rest three, which guarantee the resolution. The global outcome is a Big Crown biggest after each process, whose internal points and their links do not influence on the following points. There are two graphics files, Formation of structure and Plans, where I change the three colours by three shapes: triangle, circle and square. I recommend to see the two graphics files consecutively