Editing Graph coloring (section)

{{Short description|Methodic assignment of colors to elements of a graph}}
[[File:Petersen graph 3-coloring.svg|thumb|right|A proper vertex coloring of the [[Petersen graph]] with 3 colors, the minimum number possible.]]

In [[graph theory]], '''graph coloring''' is a methodic assignment of labels traditionally called "colors" to elements of a [[Graph (discrete mathematics)|graph]]. The assignment is subject to certain constraints, such as that no two adjacent elements have the same color. Graph coloring is a special case of [[graph labeling]]. In its simplest form, it is a way of coloring the [[Vertex (graph theory)|vertices]] of a graph such that no two adjacent vertices are of the same color; this is called a '''vertex coloring'''. Similarly, an ''[[edge coloring]]'' assigns a color to each [[Edge (graph theory)|edges]] so that no two adjacent edges are of the same color, and a '''face coloring''' of a [[planar graph]] assigns a color to each [[Face (graph theory)|face]] (or region) so that no two faces that share a boundary have the same color.

Vertex coloring is often used to introduce graph coloring problems, since other coloring problems can be transformed into a vertex coloring instance. For example, an edge coloring of a graph is just a vertex coloring of its [[line graph]], and a face coloring of a plane graph is just a vertex coloring of its [[dual graph|dual]]. However, non-vertex coloring problems are often stated and studied as-is. This is partly [[Pedagogy|pedagogical]], and partly because some problems are best studied in their non-vertex form, as in the case of edge coloring.

The convention of using colors originates from coloring the countries in a [[political map]], where each face is literally colored. This was generalized to coloring the faces of a graph [[Graph embedding|embedded]] in the plane. By planar duality it became coloring the vertices, and in this form it generalizes to all graphs. In mathematical and computer representations, it is typical to use the first few positive or non-negative integers as the "colors". In general, one can use any [[finite set]] as the "color set". The nature of the coloring problem depends on the number of colors but not on what they are.

Graph coloring enjoys many practical applications as well as theoretical challenges. Beside the classical types of problems, different limitations can also be set on the graph, or on the way a color is assigned, or even on the color itself. It has even reached popularity with the general public in the form of the popular number puzzle [[Sudoku]]. Graph coloring is still a very active field of research.

{{xref|Note: Many terms used in this article are defined in [[Glossary of graph theory]].}}