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In graph theory, graph coloring is an assignment of "colors", almost always taken to be consecutive integers starting from 1 without loss of generality, to certain objects in a graph. Such objects can be vertices, edges, faces, or a mixture of the above. Among all, vertex coloring is the most important kind, not only because it is the starting point of the entire subject, but also because other coloring problems can be transformed into a vertex version. For example, an edge coloring of a graph is just the vertex coloring of its line graph. Likewise, a face coloring of a planar graph is just the vertex coloring of its (planar) dual. However, to keep things in their perspective, non-vertex coloring problems are usually stated and studied as are.

Graph coloring is not to be confused with graph labeling , which is an assignment of labels, usually also in the form of numbers, to vertices or edges. In graph coloring problems, colors (e.g. 1, 2, 3...) are nothing more than markers for keeping track of adjacency or incidence. In graph labeling problems, labels (e.g. 1, 2, 3...) are calculable values that need to satisfy a certain numerical condition in the definition of the labeling.

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 is still a very active field of research.

Note: Many terms used in this article are defined in the glossary of graph theory.

1 Vertex coloring

When used without any qualification, a coloring of a graph is always assumed to be a vertex coloring, namely an assignment of colors to the vertices of the graph. Again, when used without any qualification, a coloring is always assumed to be proper, meaning no two adjacent vertices are assigned the same color. Here, "adjacent" means sharing the same edge. A coloring with k colors is called a (proper) k-coloring and is equivalent to the problem of partitioning the vertex set into k independent sets. The problem of coloring a graph has found a number of applications such as scheduling, register allocation in a microprocessor, frequency assignment in mobile radio s, and pattern matching.

In general, techniques for graph coloring concentrate on finding the least number of colors needed to color the graph, called its chromatic number . For example the chromatic number of a complete graph of vertices (a graph with an edge between every two vertices), is . A graph that can be assigned a (proper) k-coloring is k-colorable, and it is k-chromatic if its chormatic number is exactly k.

The problem of finding a minimum coloring of a graph is NP-hard. The corresponding decision problem(Is there a coloring which uses at most colors?) is NP-complete. There are however some efficient approximation algorithms that employ semidefinite programming .

Some properties of χ(G):

1. if and only if G is totally disconnected.
2. if and only if G has an odd cycle.
3. .
4. .
5. unless G is a clique or an odd cycle. (Brooks' theorem)
6. , for any planar graph G. This famous result, called the four-color theorem, was stated by P. J. Heawood in 1890, but remained unproven until 1976, when it was established by Kenneth Appel and Wolfgang Haken at the University of Illinois.

Here Δ(G) is the maximum degree; and ω(G), the clique number.

Comparing to finite graphs, not much about infinite graphs are known. The following is one of the few results about infinite graph coloring:

If all finite subgraphs of an infinite graph G are k-colorable, then so is G. (de Bruijn, Erdos 1951)