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Graph Coloring .

Graph Coloring is the coloring of the vertices of a graph such that no adjacent vertices share the same color.

Similarly, an edge coloring assigns colors to each edge such that no adjacent edges share the same color, and coloring of faces of a planar graph to assign a color to each face or region so that faces that share a common border have different colors.

The first result about graph coloring was only on planar graphs and coloring of maps. While trying to color a map of England Francis Guthrie postulated the theory of 4 colors, noting that 4 colors are enough to color the map so that regions sharing a common edge not receive the same color.

Labels such as red or blue are only used when the number of colors is small, and usually the colors are represented by the integers {1, 2, 3, ...}.

A coloring that uses at most k colors is called a k-coloring (proper). The smallest number of colors required to color a graph G is denoted chromatic number. A graph that can be assigned a k-coloring (proper) is k-colorable and k-chromatic if its chromatic number is exactly k. A subset of vertices assigned the same color is called a color class. Each class forms an independent set. That is, a k-coloring is the same as a partition of vertex set into k independent sets, terms and k-partite k-colorable have the same meaning.

The chromatic polynomial the number of ways in which a graph can be colored using no more than a given number of colors

A simple algorithm for graph coloring is easy to describe, but potentially very costly.
In terms of computational complexity - NP-complete What that means there is no known algorithm for optimal graph coloring is not exponential , and furthermore, if there is a non-exponential algorithm for it would not be a non-exponential solution of all problems.

A general solution to finding an optimal graph coloring is exhaustive search: start with a node, give it a color, assign colors they do not conflict with its neighbors, and so on. Try two colors, if you do not get any results, then try three and so on.

graph coloring Examples:

Petersen graph with 3 colors, commonly depicted as a pentagon with a pentagram in: