Editing Four color theorem (section)

==Formulation==
The coloring of maps can also be stated in terms of [[graph theory]], by considering it in terms of constructing a [[graph coloring]] of the [[planar graph]] of adjacencies between regions. In graph-theoretic terms, the theorem states that for a [[Loop (graph theory)|loopless]] planar graph <math>G</math>, its [[chromatic number]] is <math>\chi(G) \leq 4</math>.

For this to be meaningful, the intuitive statement of the four color theorem – "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.

First, regions are adjacent if they share a boundary segment; two regions that share only isolated boundary points are not considered adjacent. (Otherwise, a map in a shape of a [[pie chart]] would make an arbitrarily large number of regions 'adjacent' to each other at a common corner, and require an arbitrarily large number of colors as a result.) Second, bizarre regions, such as those with finite area but infinitely long perimeter, are not allowed; maps with such regions can require more than four colors.{{sfnp|Hudson|2003}} (To be safe, we can restrict to regions whose boundaries consist of finitely many straight line segments. It is allowed that a region has [[Enclave and exclave|enclaves]], that is it entirely surrounds one or more other regions.) Note that the notion of "contiguous region" (technically: [[Connected space|connected]] [[Open set|open]] subset of the plane) is not the same as that of a "country" on regular maps, since countries need not be contiguous (they may have [[Enclave and exclave|exclaves]]; e.g., the [[Cabinda Province]] as part of [[Angola]], [[Nakhchivan Autonomous Republic|Nakhchivan]] as part of [[Azerbaijan]], [[Kaliningrad Oblast|Kaliningrad]] as part of Russia, [[France]] with its [[Overseas France|overseas territories]], and [[Alaska]] as part of the [[United States]] are not contiguous). If we required the entire territory of a country to receive the same color, then four colors are not always sufficient. For instance, consider a simplified map:

[[File:4CT Inadequacy Example.svg|center|200px]]

In this map, the two regions labeled ''A'' belong to the same country. 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.

[[File:Four Colour Planar Graph.svg|thumb|right|A map with four regions, and the corresponding planar graph with four vertices.]]
A simpler statement of the theorem uses [[graph theory]]. The set of regions of a map can be represented more abstractly as an [[undirected graph]] that has a [[vertex (graph theory)|vertex]] for each region and an [[edge (graph theory)|edge]] for every pair of regions that share a boundary segment. This graph is [[planar graph|planar]]: it can be drawn in the plane without crossings by placing each vertex at an arbitrarily chosen location within the region to which it corresponds, and by drawing the edges as curves without crossings that lead from one region's vertex, across a shared boundary segment, to an adjacent region's vertex. Conversely any planar graph can be formed from a map in this way. In graph-theoretic terminology, the four-color theorem states that the vertices of every planar graph can be colored with at most four colors so that no two adjacent vertices receive the same color, or for short: every planar graph is [[Graph coloring|four-colorable]].<ref>{{harvtxt|Thomas|1998|p=849}}; {{harvtxt|Wilson|2014}}).</ref>