Graph coloring

In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called colors to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices share the same color; this is called a vertex coloring. 

Similarly, an edge coloring assigns a color to each edge so that no two adjacent edges share the same color, and a face coloring of a planar graph assigns a color to each face or region so that no two faces that share a boundary have the same color. Vertex coloring is the other coloring problems can be transformed into a vertex version.

For example, an edge coloring of a graph is just a vertex coloring of its line graph, and a face coloring of a planar graph is just a vertex coloring of its planar dual. However, non-vertex coloring problems are often stated and studied as is. That is partly for perspective are best studied in non-vertex form, as for instance is edge coloring.

The convention of using colors originates from coloring where each face is literally colored. This was generalized to coloring the faces of a graph embedded in the plane. By planar duality and in this form it generalizes to all graphs. The nature of the coloring problem depends on the number of colors but not on what they are.