A 3-coloring suits this graph, but fewer colors would result in adjacent vertices of the same color. Finding the minimum number of colors is NP-hard.

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. A graph with a 3-coloring File links The following pages link to this file: Graph coloring Categories: GFDL images ... In computational complexity theory, NP-hard (Non-deterministic Polynomial-time hard) refers to the class of decision problems that contains all problems H such that for all decision problems L in NP there is a polynomial-time many-one reduction to H. Informally this class can be described as containing... In mathematics and computer science, graph theory studies the properties of graphs. ... In graph theory, the line graph L(G) of a graph G is a graph such that each vertex of L(G) represents an edge of G; and any two vertices of L(G) are adjacent if and only if their corresponding edges are incident, meaning they share a common... One major problem that has plagued graph theory since its inception is the lack of consistency in terminology. ... One major problem that has plagued graph theory since its inception is the lack of consistency in terminology. ...

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. In the mathematical discipline of graph theory, a graph labeling is the assignment of unique identifiers to the edges and vertices of a graph. ...

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. A difficult Sudoku puzzle. ...

Note: Many terms used in this article are defined in the glossary of graph theory. One major problem that has plagued graph theory since its inception is the lack of consistency in terminology. ...

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 radios, and pattern matching. One major problem that has plagued graph theory since its inception is the lack of consistency in terminology. ... One major problem that has plagued graph theory since its inception is the lack of consistency in terminology. ... In graph theory, an independent, or stable, set in a graph G, which contains vertices V, is a set of vertices V (a subset of V) such that for every two vertices in V, there is no edge connecting the two. ... For schedule in computer science, see schedule (computer science). ... In computer architecture, a processor register is a small amount of very fast computer memory used to speed the execution of computer programs by providing quick access to commonly used values—typically, the values being in the midst of a calculation at a given point in time. ... Microprocessors, including an Intel 80486DX2 and an Intel 80386 A microprocessor (abbreviated as µP or uP) is an electronic computer central processing unit (CPU) made from miniaturized transistors and other circuit elements on a single semiconductor integrated circuit (IC) (aka microchip or just chip). ... In telecommunication, the term frequency assignment has the following meanings: Authorization, given by an Administration, for a radio station to use a radio frequency or radio frequency channel under specified conditions. ... Pattern matching is a feature in functional programming languages such as Mathematica, Haskell and ML. With certain data types implemented in these programming languages, such as a list, the construction of these types form a very definite structure. ...

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

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 k colors?) is NP-complete, and in fact was one of Karp's 21 NP-complete problems. It remains NP-complete even on planar graphs of degree at most 4, as shown by Garey and Johnson in 1974, although on planar graphs it's trivial for k ≠ 3 (due to the four color theorem). There are however some efficient approximation algorithms that employ semidefinite programming. In computational complexity theory, NP-hard (Non-deterministic Polynomial-time hard) refers to the class of decision problems that contains all problems H such that for all decision problems L in NP there is a polynomial-time many-one reduction to H. Informally this class can be described as containing... In logic, a decision problem is determining whether or not there exists a decision procedure or algorithm for a class S of questions requiring a Boolean value (i. ... In complexity theory, the NP-complete problems are the most difficult problems in NP, in the sense that they are the ones most likely not to be in P. The reason is that if you could find a way to solve an NP-complete problem quickly, then you could use... It was in 1971 that the first NP-complete problem, the boolean satisfiability problem, was discovered and proven to be NP-complete by Stephen Cook. ... Example of a four color map The four color theorem states that any plane separated into regions, such as a political map of the counties of a state, can be colored using no more than four colors in such a way that no two adjacent regions receive the same color. ...

Some properties of χ(G):

1. χ(G) = 1 if and only if G is totally disconnected.
2. χ(G) ≥ 3 if and only if G has an odd cycle.
3. χ(G) ≥ ω(G).
4. χ(G) ≤ Δ(G)+1.
5. χ(G) ≤ Δ(G) unless G is a complete graph or an odd cycle (Brooks' theorem).
6. χ(G) ≤ 4, 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. One major problem that has plagued graph theory since its inception is the lack of consistency in terminology. ... In the mathematical field of graph theory a cycle graph or circle graph is a graph that consists of a cycle. ... One major problem that has plagued graph theory since its inception is the lack of consistency in terminology. ... In graph theory, a planar graph is a graph that can be embedded in a plane so that no edges intersect. ... Example of a four color map The four color theorem states that every possible geographical map can be colored with at most four colors in such a way that no two adjacent regions receive the same colour. ... Kenneth Appel is a mathematician who, in 1976 with colleague Wolfgang Haken at the University of Illinois in Urbana, solved one of the most famous problems in mathematics, the four-color theorem. ... Wolfgang Haken (born June 21, 1928) is a mathematician who specialized in topology, in particular 3-manifolds. ... The University of Illinois is the set of three public universities in Illinois. ... One major problem that has plagued graph theory since its inception is the lack of consistency in terminology. ... One major problem that has plagued graph theory since its inception is the lack of consistency in terminology. ...

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, Erdős 1951)

A method to calculate the chromatic number is using the chromatic polynomial χ_G(x), which encodes all possible vertex colorings with x colors for a given graph G. One major problem that has plagued graph theory since its inception is the lack of consistency in terminology. ... In the mathematical field of graph theory the chromatic polynomial for a given graph is a polynomial which encodes the number of different ways to vertex color the graph using n colors. ...

Unfortunately the polynomial is quite hard to construct for an arbitrary graph.

List of other colorings

Not only can the idea of vertex coloring be extended to edges, but also be added with different conditions to form new structures and problems.

• edge coloring - edges are colored
• list coloring - each vertex chooses from a list of colors
• list edge-coloring - each edge chooses from a list of colors
• total coloring - vertices and edges are colored
• harmonious coloring - every pair of colors appears on at most one edge
• complete coloring - every pair of colors appears on at least one edge
• exact coloring - every pair of colors appears on exactly one edge
• acyclic coloring - every 2-chromatic subgraph is acyclic
• strong coloring - every color appears in every partition of equal size exactly once
• on-line coloring - the instance of the problem is not given in advance and its successive parts become known over time
• equitable coloring - the sizes of color classes differ by at most one
• sum-coloring - the criterion of minimalization is the sum of colors
• T-coloring - distance between two colors of adjacent vertices must not belong to fixed set T
• rank coloring - if two vertices have the same color i, then every path between them contain a vertex with color greater than i.
• interval edge-coloring - a color of edges meeting in a common vertex must be contiguous.
• circular coloring - motivated by task systems in which production proceeds in a cyclic way
• path coloring - models a routing problem in graphs
• fractional coloring - vertices may have multiple colors, and on each edge the sum of the color parts of each vertex is not greater than one

Some improper colorings: In graph theory, similar to its vertex counterpart, an edge coloring of a graph, when used without any qualification, is always assumed to be a proper coloring on the edges, meaning no two incident edges are assigned the same color. ... In graph theory, list coloring is a type of graph coloring. ... In mathematics, list edge-coloring is a type of graph coloring. ... In graph theory, total coloring is a type of coloring on the vertices and edges of a graph. ... In graph theory, a harmonious coloring is a (proper) vertex coloring in which every pair of colors appears on at most one pair of adjacent vertices. ... In graph theory, complete coloring is the opposite of harmonious coloring in the sense that it is a vertex coloring in which every pair of colors appears on at least one pair of adjacent vertices. ... In graph theory, an exact coloring is a (proper) vertex coloring in which every pair of colors appears on exactly one pair of adjacent vertices. ... In graph theory, an acyclic coloring is a (proper) vertex coloring in which every 2_chromatic subgraph is acyclic. ... In graph theory, a strong coloring, with respect to a partition of the vertices into (disjoint) subsets of equal sizes, is a (proper) vertex coloring in which every color appears exactly once in every partition. ... Fractional coloring is a topic in a young branch of graph theory known as fractional graph theory. ...

• cocoloring -- every color class induces an independent set or a clique
• subcoloring -- every color class induces a union of cliques

In graph theory, a cocoloring of a graph G is an assignment of colors to the vertices such that each color class forms an independent set in G or the complement of G. The cochromatic number z(G) of G is the least number of colors needed in any cocolorings... In graph theory, a subcoloring is an assignment of colors to a graphs vertex such that each color class induces a vertex disjoint union of cliques. ...

Literature

• R. L. Brooks: On colouring the nodes of a network. In: Proc. Cambridge Phil. Soc. 37, 1941, 194–197.
• N. G. de Bruijn, P. Erdős: A colour problem for infinite graphs and a problem in the theory of relations. In: Nederl. Akad. Wetensch. Proc. Ser. A 54, 1951 371–373. (Indag. Math. 13.)
• Tommy R. Jensen, Bjarne Toft: Graph coloring problems. Wiley-Interscience, New York 1995 ISBN 0-471-02865-7.
• Marek Kubale and others: Graph Colorings. American Mathematical Society 2004 ISBN 0-8218-3458-4.
• M. R. Garey, D. S. Johnson, and L. Stockmeyer. Some simplified NP-complete problems (http://portal.acm.org/citation.cfm?id=803884). Proceedings of the sixth annual ACM symposium on Theory of computing, p.47-63. 1974.

See also

• Four-color theorem
• Uniquely colorable graph
• Critical graph
• Graph homomorphism
• Perfect graph

Example of a four color map The four color theorem states that every possible geographical map can be colored with at most four colors in such a way that no two adjacent regions receive the same colour. ... In graph theory, a uniquely colorable graph is a k-chromatic graph that has only one possible (proper) k-coloring up to permutation of the colors. ... In general the notion of criticality can refer to any measure. ... In the mathematical field of graph theory a graph homomorphism is a mapping between two graphs that respects their structure. ... In graph theory, a perfect graph is a graph in which the chromatic number of every induced subgraph equals the clique number of that subgraph. ...

External links

• Graph Coloring Page (http://www.cs.ualberta.ca/~joe/Coloring/index.html) by Joseph Culberson (graph coloring programs)