# Graph coloring 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.

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.

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## 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.

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.

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 [itex]k[itex] 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.

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.

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 χG(x), which encodes all possible vertex colorings with x colors for a given graph G.

[itex]\chi(G) = \min \lbrace x \in \mathbb{N} | \chi_G(x) \ge 0 \rbrace[itex]

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:

• cocoloring -- every color class induces an independent set or a clique
• subcoloring -- every color class induces a 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.

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