Forcing two separate regions to have the same color can be modelled by adding a ‘handle’ joining them outside the plane. Such construction makes the problem equivalent to coloring a map on a torus (a surface of genus 1), which requires up to 7 colors for an arbitrary map.
Why do you need five colors to get the same color?
If we wanted those regions to receive the same color, then five colors would be required, since the two A regions together are adjacent to four other regions, each of which is adjacent to all the others. A similar construction also applies if a single color is used for all bodies of water, as is usual on real maps.
When to use the four color theorem in graph theory?
For maps in which more than one country may have multiple disconnected regions, six or more colors might be required. A map with four regions, and the corresponding planar graph with four vertices. A simpler statement of the theorem uses graph theory.
What is the difference between a simple map and a planar map?
From Gonthier (2008): “Definitions: A planar map is a set of pairwise disjoint subsets of the plane, called regions. A simple map is one whose regions are connected open sets. Two regions of a map are adjacent if their respective closures have a common point that is not a corner of the map.
Do you have to have five colors to get the same color?
If we wanted those regions to receive the same color, then five colors would be required, since the two A regions together are adjacent to four other regions, each of which is adjacent to all the others.
Which is the correct statement of the four color theorem?
{\\displaystyle \\chi (G^ {*})\\leq 4} . The intuitive statement of the four color theorem, i.e. “given any separation of a plane into contiguous regions, the regions can be colored using at most four colors so that no two adjacent regions have the same color”, needs to be interpreted appropriately to be correct.
