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

In graph theory, graph coloring is an assignment of "colors" to certain objects in 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, called a vertex coloring. Similarly, an edge coloring assigns a color to each edge so that no two incident 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 share the same color.

Vertex coloring is the starting point of the subject, and 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, and 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.

The convention of using colors comes from graph drawings of graph colorings, where each node or edge is literally colored to indicate its mapping. In computer representations it's more typical to use nonnegative integers, and in general any mapping from the graph objects into a finite set can be used.

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 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.

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

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 nearly always assumed to be proper, meaning no two adjacent vertices are assigned the same color. Here, "adjacent" means sharing the same edge. A coloring using at most k colors is called a (proper) k-coloring and is equivalent to the problem of partitioning the vertex set into k or fewer independent sets. The problem of coloring a graph has found a number of applications such as scheduling, register allocation in compilers, frequency assignment in mobile radios, and pattern matching.

The least number of colors needed to color the graph is called its chromatic number χ. For example the chromatic number of a complete graph K_n of n vertices (a graph with an edge between every two vertices), is χ(K_n) = 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.

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^[1], 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.

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 (equivalently, if G is not bipartite).
3. χ(G) ≥ ω(G).
4. χ(G) ≤ Δ(G)+1.
5. χ(G) ≤ Δ(G) for connected 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 (first written reference by Augustus De Morgan in 1852), but remained unproven until 1976, when it was established by Kenneth Appel and Wolfgang Haken at the University of Illinois at Urbana-Champaign.

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

About infinite graphs, much less is 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)

For planar graphs, vertex coloring is dual to nowhere-zero flows. The chromatic number of the plane is unknown, although it is one of 4, 5, 6, or 7.

Vertex coloring in the general case is an NP-Complete problem. Instead of asking for the smallest number of colors needed to color the graph, we can ask easier questions of the form "Can we color the graph with at most k colors?"

The case k = 2 is equivalent to determining whether or not the graph is bipartite. This can be accomplished in polynomial time. For k >= 3 the problem is NP-Complete. By the gap theorem, this implies that the problem can not be approximated by a polynomial algorithm within a factor of 4/3 unless P=NP.

A remarkable result of László Lovász states that it is possible to determine the chromatic number of a perfect graph in polynomial time.

The coloring algorithms can be divided into two categories:

- The optimal coloring algorithms (for example the algorithm of Zykov, the method of branch and bound, etc.).

- The algorithms that do not ensure a result with the smallest possible number of colors. Here we can find the sequential algorithms (those that color one vertex at a time), heuristic algorithms, global randomized algorithms, metaheuristic algorithms (using simulated annealing, tabu search, etc.), and genetic algorithms, to name several types.

The Welsh-Powell algorithm for coloring a graph uses a simple heuristic improvement to a completely naive greedy algorithm. It is easy to show that this approach uses at most Δ(G) + 1 colors, where Δ(G) is the maximum degree of the graph. The algorithm is as follows:

1. Sort the vertices in decreasing order of degree and initially have every vertex uncolored.
2. Traverse the vertices in order, giving a vertex color 1 if it is uncolored and it does not yet have a neighbor with color 1.
3. Repeat this process with colors 2, 3, etc. until no vertex is uncolored.

This algorithm does not necessarily find a χ(G) coloring.

This graph can be 3-colored in 12 different ways.

The chromatic polynomial counts the number of ways a graph can be colored using no more than a given number of colors. For example, using three colors, the graph in the image to the right can be colored in 12 ways. With only two colors, it cannot be colored at all. With four colors, it can be colored in 24 + 4⋅12 = 72 ways: using all four colors, there are 4! = 24 valid colorings (every assignment of four colors to any 4-vertex graph is a proper coloring); and for every choice of three of the four colors, there are 12 valid 3-colorings. So, for the graph in the example, a table of the number of valid colorings would start like this:

The chromatic polynomial is a function P(G,t) that counts the number of t-colorings of G. As the name indicates, for a given G the function is indeed a polynomial in t. For the example graph, P(G,t) = t(t − 1)²(t − 2), and indeed P(G,4) = 72.

The chromatic polynomial includes at least as much information about the colorability of G as does the chromatic number. Indeed, χ is the smallest positive integer that is not a root of the chromatic polynomial,

χ(G) = min{k:P(G,k) > 0}

It was first used by Birkhoff and Lewis in their attack on the four-color theorem. It remains an unsolved problem to characterize graphs which have the same chromatic polynomial and to determine precisely which polynomials are chromatic.

The Petersen graph has chromatic number 3.

• P(G,0) = 0
• P(G,1) = 0 if G contains an edge
• P(G,t) = 0, if t < χ(G).
• ( − 1) ^{| V(G) |} P(G, − 1) is the number of acyclic orientations of G^[2]
• If G has n vertices, m edges, and k components G₁,G₂,…,G_k, then
  • P(G,t) has degree n.
  • The coefficient of tⁿ in P(G,t) is 1.
  • The coefficient of t^{n − 1} in P(G,t) is − m.
  • The coefficients of t⁰,t¹, … t^{k − 1} are all zero.
  • The coefficient of t^k is nonzero.
  • P(G,t) = P(G₁,t)P(G₂,t)⋯P(G_k,t)
• The coefficients of every chromatic polynomial alternate in signs.
• A graph G with n vertices is a tree if and only if P(G,t) = t(t − 1)^{n − 1}.
• The derivative evaluated at unity, P'(G,1) is the chromatic invariant θ(G) up to sign

Whenever G contains a loop, it cannot be legally colored at all, so:

P(G,t) = 0.

If e is not a loop, then the chromatic polynomial satisfies the recurrence relation:

P(G,t) = P(G − e,t) − P(G / e,t)

where G − e is the graph with the edge removed, and G / e is the graph with the edge's endpoints contracted into a single vertex.

The two expressions give rise to a recursive procedure, called the deletion–contraction algorithm. In the recursive step, the instance is transformed into two instances that are at least one edge smaller, so the running time comes within a polynomial factor of 2^m, where m is the number of edges. The analysis can be improved to 1.62^{n + m}.

Other algorithms are known, but all are exponential in the size of the graph. For some classes of graphs, polynomial-time algorithms are known. These include the empty and complete graphs, forests, chordal graphs, wheels, and ladders. In general, computing the chromatic polynomial is #P-complete, so it is unlikely that a polynomial time algorithm for all graphs will be found.

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.

Some improper colorings:

• Brooks, R. L. (1941). "On colouring the nodes of a network". Proc. Cambridge Phil. Soc. 37: 194–197. 
• de Bruijn, N. G. & Erdős, P. (1951), "A colour problem for infinite graphs and a problem in the theory of relations", Nederl. Akad. Wetensch. Proc. Ser. A 54: 371–373 (Indag. Math. 13.)
• Jensen, Tommy R. & Toft, Bjarne (1995), Graph coloring problems, Wiley-Interscience, New York, ISBN 0-471-02865-7
• Marek Kubale and others (2004). Graph Colorings. American Mathematical Society. ISBN 0-8218-3458-4. 
• Michael R. Garey, David S. Johnson, and L. Stockmeyer (1974). "Some simplified NP-complete problems". Proceedings of the sixth annual ACM symposium on Theory of computing: 47-63. 
• Michael R. Garey and David S. Johnson (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman. ISBN 0-7167-1045-5.  A1.1: GT4, pg.191.

Wikimedia Commons has media related to:

Graph coloring

• Four-color theorem
• Uniquely colorable graph
• Critical graph
• Graph homomorphism
• Perfect graph
• Hadwiger–Nelson problem
• Sudoku

• Chromatic number of a space at PlanetMath.org
• Graph Coloring Page by Joseph Culberson (graph coloring programs)
• CoLoRaTiOn by Jim Andrews and Mike Fellows is a graph coloring puzzle
• GF-Graph Coloring Program [1]
• Links to Graph Coloring source codes