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 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 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, and partly because some problems are best studied in non-vertex form, as for instance is edge coloring.
The convention of using colors originates from coloring the countries of a map, where each face is literally colored. This was generalized to coloring the faces of a graph embedded in the plane. By planar duality it became coloring the vertices, and in this form it generalizes to all graphs. In mathematical and computer representations it is typical to use the first few positive or nonnegative integers as the "colors". In general one can use any finite set as the "color set". The nature of the coloring problem depends on the number of colors but not on what they are.
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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The first results about graph coloring deal almost exclusively with planar graphs in the form of the coloring of maps. While trying to color a map of the counties of England, Francis Guthrie postulated the four color conjecture, noting that four colors were sufficient to color the map so that no regions sharing a common border received the same color. Guthrie’s brother passed on the question to his mathematics teacher Augustus de Morgan at University College, who mentioned it in a letter to William Hamilton in 1852. Arthur Cayley raised the problem at a meeting of the London Mathematical Society in 1879. Over a century of failed attempts to establish the conjecture followed, until the four colour theorem was finally proved in 1976, a result that is also noteworthy for being the first major computer-aided proof.
In 1912, Birkhoff introduced the chromatic polynomial to study the coloring problems, which was generalised to the Tutte polynomial by Tutte, important structures in algebraic graph theory. Kempe had already drawn attention to the general, non-planar case in 1879[1], and many results on generalisations of planar graph coloring to surfaces of higher order followed in the early 20th century.
In 1960, Claude Berge formulated another conjecture about graph coloring, the strong perfect graph conjecture, originally motivated by an information-theoretic concept called the zero-error capacity of a graph introduced by Shannon. The conjecture remained unresolved for 40 years, until it was established as the celebrated strong perfect graph theorem in 2002 by Chudnovsky, Robertson, Seymour, Thomas 2002.
Graph coloring has been studied as an algorithmic problem since the early 1970s. The chromatic number problem is one of Karp’s 21 NP-complete problems from 1972, around the time of various exponential-time algorithms based on backtracking and Zykov’s deletion--contraction recurrence from 1949[2]. One of the major applications of graph coloring, register allocation in compilers was introduced in 1981.
When used without any qualification, a coloring of a graph is almost always a proper vertex coloring, namely an assignment of colors to the graph’s vertices such that no two vertices sharing the same edge (graph theory) are assigned the same color. A coloring using at most k colors is called a (proper) k-coloring. A subset of vertices assigned to the same color is called a color class, every such class forms an independent set. Thus, a k-coloring is the same as a partition the vertex set into k independent sets.
The smallest number of colors needed to color a graph G is called its chromatic number, χ(G). 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 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:
Available colors | 1 | 2 | 3 | 4 | … |
Number of colorings | 0 | 0 | 12 | 72 | … |
The chromatic polynomial is a function that counts the number of
-colorings of
. As the name indicates, for a given
the function is indeed a polynomial in
. For the example graph,
, and indeed
.
The chromatic polynomial includes at least as much information about the colorability of as does the chromatic number. Indeed, χ is the smallest positive integer that is not a root of the chromatic polynomial,
It was first used by Birkhoff and Lewis in their attack on the four-color theorem. They conjectured that, for a planar graph ,
has no zeros in the region
. Although it is known that such a chromatic polynomial has no zeros in the region
and that
. Their conjecture is still unresolved.
It remains an unsolved problem to characterize graphs which have the same chromatic polynomial and to determine precisely which polynomials are chromatic.
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Cycle ![]() |
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An edge coloring of a graph, is a proper coloring of the edges, meaning an assignment of colors to edges so that no vertex is incident to two edges of the same color. An edge coloring with k colors is called a k-edge-coloring and is equivalent to the problem of partitioning the edge set into k matchings. The smallest number of colors needed for an edge coloring of a graph G is the chromatic index, or edge chromatic number, χ′(G).
Coloring can also be considered for signed graphs and gain graphs.
Every n-vertex graph is n-colorable, and the complete graph of n vertices, requires
colors.
For planar graphs, vertex colorings are essentially dual to nowhere-zero flows.
About infinite graphs, much less is known. The following is one of the few results about infinite graph coloring:
The chromatic number of the plane, where two points are adjacent if they have unit distance, is unknown, although it is one of 4, 5, 6, or 7. Other open problems concerning the chromatic number of graphs include the Hadwiger conjecture stating that every graph with chromatic number k has a complete graph on k vertices as a minor, and the Erdős–Faber–Lovász conjecture bounding the chromatic number of unions of complete graphs that have at exactly one vertex in common to each pair.
Graph coloring is computationally hard. It is NP-complete to decide if a given graph admits a coloring which uses at most a given number of colors. The problem remains NP-complete even on planar graphs of degree at most 4 [5].
Determining if a graph can be colored with colors is equivalent to determining whether or not the graph is bipartite, and thus polynomial time-computable. As a consequence of the Strong Perfect Graph Theorem, the chromatic number of a perfect graph can be computed in polynomial time. Other classes of graphs that admit polynomial time algorithms are forests and chordal graphs.
Brute-force search for a -coloring considers every of the
assignments of colors to vertices and checks for each if it is legal. To compute the chromatic number and the chromatic polynomial, this procedure is used for every
, for a total running time of
, impractical for all but the smallest input graphs.
This can be improved based on Zykov’s identity: If is not a loop, then the chromatic number satisfies the recurrence relation:
where and
are nonadjecent vertices,
is the graph with the edge
added, and
is the graph where the two vertices are contracted into a single vertex.
The expressions gives rise to a recursive procedure, called the deletion–contraction algorithm. The running time satisfies the same recurrence relation as the Fibonacci numbers, so in the worst case, the algorithm runs in time within a polynomial factor of [6]. The analysis can be improved to within a polynomial factor of the number
of spanning trees of the input graph [7]. In practice, branch and bound strategies and graph isomorphism rejection are employed to avoid some recursive calls, the running time depends on the heuristic used to pick the vertex pair.
The greedy algorithm considers the vertices in a specific order and assigns to
the smallest available color not used by
’s neighbours among
, adding a fresh color if needed. The quality of the resulting coloring depends on the chosen ordering. There exists an ordering that leads to a greedy coloring with the optimal number of
colors. On the other hand, greedy colorings can be arbitrarily bad; for example, there are graphs that can be 2-colored, but where there is an ordering that leads to a greedy coloring with
colors.
If the vertices are ordered according to their degrees, the resulting greedy coloring uses at most colors, at most one more than the graph’s maximum degree. This heuristic is sometimes called the Welsh–Powell algorithm[8]. Another heuristic establishes the ordering dynamically while the algorithm proceeds, choosing next the vertex with the largest number of different colors[9]. Many other graph coloring heuristics are similarly based on greedy coloring for a specific static or dynamic strategy of ordering the vertices, these algorithms are sometimes called sequential coloring algorithms.
Graph coloring belongs to the hardest problems in NP in the sense that even approximating the correct answer is difficult.
The best known approximation algorithm computes a coloring of size at most within a factor of the chromatic number.[10] For all
it is NP-hard to approximate the chromatic number within
[11]
Other algorithms are known, but all are exponential in the size of the graph. In general, computing the chromatic polynomial is #P-complete, so it is unlikely that a polynomial time algorithm for all graphs will be found.
The problem of coloring a graph has found a number of applications. Some of them are scheduling, register allocation in compilers, bandwidth allocation, and pattern matching.
The recreational puzzle Sudoku can be seen as completing a 9-coloring on given specific graph with 81 vertices.