
In
graph theory
In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ...
, graph coloring is a methodic assignment of labels traditionally called "colors" to elements of a
graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discret ...
. The assignment is subject to certain constraints, such as that no two adjacent elements have the same color. Graph coloring is a special case of
graph labeling
In the mathematical discipline of graph theory, a graph labeling is the assignment of labels, traditionally represented by integers, to edges and/or vertices of a graph.
Formally, given a graph , a vertex labeling is a function of to a set ...
. In its simplest form, it is a way of coloring the
vertices of a graph such that no two adjacent vertices are of the same color; this is called a vertex coloring. Similarly, an ''
edge coloring'' assigns a color to each
edges so that no two adjacent edges are of the same color, and a face coloring of a
planar graph
In graph theory, a planar graph is a graph (discrete mathematics), graph that can be graph embedding, embedded in the plane (geometry), plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. ...
assigns a color to each
face
The face is the front of the head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affect th ...
(or region) so that no two faces that share a boundary have the same color.
Vertex coloring is often used to introduce graph coloring problems, since other coloring problems can be transformed into a vertex coloring instance. For example, an edge coloring of a graph is just a vertex coloring of its
line graph
In the mathematics, mathematical discipline of graph theory, the line graph of an undirected graph is another graph that represents the adjacencies between edge (graph theory), edges of . is constructed in the following way: for each edge i ...
, and a face coloring of a plane graph is just a vertex coloring of its
dual. However, non-vertex coloring problems are often stated and studied as-is. This is partly
pedagogical
Pedagogy (), most commonly understood as the approach to teaching, is the theory and practice of learning, and how this process influences, and is influenced by, the social, political, and psychological development of learners. Pedagogy, taken ...
, and partly because some problems are best studied in their non-vertex form, as in the case of edge coloring.
The convention of using colors originates from coloring the countries in a
political 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 non-negative integers as the "colors". In general, one can use any
finite set
In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example,
is a finite set with five elements. Th ...
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
Sudoku (; ; originally called Number Place) is a logic puzzle, logic-based, combinatorics, combinatorial number-placement puzzle. In classic Sudoku, the objective is to fill a 9 × 9 grid with digits so that each column, each row, and ...
. Graph coloring is still a very active field of research.
History

The first results about graph coloring deal almost exclusively with
planar graphs
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cro ...
in the form of
map coloring.
While trying to color a map of the counties of England,
Francis Guthrie
Francis Guthrie (born 22 January 1831 in London; d. 19 October 1899 in Claremont, Cape Town) was a Cape Colony mathematician and botanist who first posed the Four Colour Problem in 1852. He studied mathematics under Augustus De Morgan, and bo ...
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
Augustus De Morgan (27 June 1806 – 18 March 1871) was a British mathematician and logician. He is best known for De Morgan's laws, relating logical conjunction, disjunction, and negation, and for coining the term "mathematical induction", the ...
at
University College
In a number of countries, a university college is a college institution that provides tertiary education but does not have full or independent university status. A university college is often part of a larger university. The precise usage varies f ...
, who mentioned it in a letter to
William Hamilton in 1852.
Arthur Cayley
Arthur Cayley (; 16 August 1821 – 26 January 1895) was a British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics, and was a professor at Trinity College, Cambridge for 35 years.
He ...
raised the problem at a meeting of the
London Mathematical Society
The London Mathematical Society (LMS) is one of the United Kingdom's Learned society, learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh ...
in 1879. The same year,
Alfred Kempe
Sir Alfred Bray Kempe FRS (6 July 1849 – 21 April 1922) was a mathematician best known for his work on linkages and the four colour theorem.
Biography
Kempe was the son of the Rector of St James's Church, Piccadilly, the Rev. John Edwar ...
published a paper that claimed to establish the result, and for a decade the four color problem was considered solved. For his accomplishment Kempe was elected a Fellow of the
Royal Society
The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national academy of sciences. The society fulfils a number of roles: promoting science and its benefits, re ...
and later President of the London Mathematical Society.
In 1890,
Percy John Heawood pointed out that Kempe's argument was wrong. However, in that paper he proved the
five color theorem, saying that every planar map can be colored with no more than ''five'' colors, using ideas of Kempe. In the following century, a vast amount of work was done and theories were developed to reduce the number of colors to four, until the four color theorem was finally proved in 1976 by
Kenneth Appel and
Wolfgang Haken. The proof went back to the ideas of Heawood and Kempe and largely disregarded the intervening developments.
The proof of the four color theorem is noteworthy, aside from its solution of a century-old problem, for being the first major computer-aided proof.
In 1912,
George David Birkhoff
George David Birkhoff (March21, 1884November12, 1944) was one of the top American mathematicians of his generation. He made valuable contributions to the theory of differential equations, dynamical systems, the four-color problem, the three-body ...
introduced the
chromatic polynomial to study the coloring problem, which was generalised to the
Tutte polynomial by
W. T. Tutte, both of which are important invariants in
algebraic graph theory. Kempe had already drawn attention to the general, non-planar case in 1879, 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 by
Chudnovsky,
Robertson,
Seymour, and
Thomas
Thomas may refer to:
People
* List of people with given name Thomas
* Thomas (name)
* Thomas (surname)
* Saint Thomas (disambiguation)
* Thomas Aquinas (1225–1274) Italian Dominican friar, philosopher, and Doctor of the Church
* Thomas the A ...
in 2002.
Graph coloring has been studied as an algorithmic problem since the early 1970s: the chromatic number problem (see section ' below) is one of
Karp's 21 NP-complete problems from 1972, and at approximately the same time various exponential-time algorithms were developed based on backtracking and on the deletion-contraction recurrence of . One of the major applications of graph coloring,
register allocation
In compiler optimization, register allocation is the process of assigning local automatic variables and Expression (computer science), expression results to a limited number of processor registers.
Register allocation can happen over a basic bloc ...
in compilers, was introduced in 1981.
Definition and terminology
Vertex coloring
When used without any qualification, a coloring of a graph almost always refers to a ''proper vertex coloring'', namely a labeling of the graph's vertices with colors such that no two vertices sharing the same
edge have the same color. Since a vertex with a
loop (i.e. a connection directly back to itself) could never be properly colored, it is understood that graphs in this context are loopless.
The terminology of using ''colors'' for vertex labels goes back to map coloring. Labels like ''red'' and ''blue'' are only used when the number of colors is small, and normally it is understood that the labels are drawn from the
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s .
A coloring using at most colors is called a (proper) -coloring. The smallest number of colors needed to color a graph is called its chromatic number, and is often denoted . Sometimes is used, since is also used to denote the
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's ...
of a graph. A graph that can be assigned a (proper) -coloring is -colorable, and it is -chromatic if its chromatic number is exactly . A subset of vertices assigned to the same color is called a ''color class''; every such class forms an
independent set. Thus, a -coloring is the same as a partition of the vertex set into independent sets, and the terms ''-partite'' and ''-colorable'' have the same meaning.
Chromatic polynomial

The chromatic polynomial counts the number of ways a graph can be colored using some of a given number of colors. For example, using three colors, the graph in the adjacent image 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 that counts the number of -colorings of . As the name indicates, for a given the function is indeed a
polynomial
In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
in . For the example graph, , and indeed .
The chromatic polynomial includes more information about the colorability of than does the chromatic number. Indeed, is the smallest positive integer that is not a zero of the chromatic polynomial .
Edge coloring
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 colors is called a -edge-coloring and is equivalent to the problem of partitioning the edge set into
matchings. The smallest number of colors needed for an edge coloring of a graph is the chromatic index, or edge chromatic number, . A Tait coloring is a 3-edge coloring of a
cubic graph. The
four color theorem
In mathematics, the four color theorem, or the four color map theorem, states that no more than four colors are required to color the regions of any map so that no two adjacent regions have the same color. ''Adjacent'' means that two regions shar ...
is equivalent to the assertion that every planar cubic
bridgeless graph admits a Tait coloring.
Total coloring
Total coloring is a type of coloring on the vertices ''and'' edges of a graph. When used without any qualification, a total coloring is always assumed to be proper in the sense that no adjacent vertices, no adjacent edges, and no edge and its end-vertices are assigned the same color. The total chromatic number of a graph is the fewest colors needed in any total coloring of .
Face coloring
For a graph with a strong embedding on a surface, the face coloring is the dual of the vertex coloring problem.
Tutte's flow theory
For a graph ''G'' with a strong embedding on an orientable surface,
William T. Tutte discovered that if the graph is ''k''-face-colorable then ''G'' admits a nowhere-zero ''k''-flow. The equivalence holds if the surface is sphere.
Unlabeled coloring
An unlabeled coloring of a graph is an
orbit
In celestial mechanics, an orbit (also known as orbital revolution) is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an ...
of a coloring under the action of the
automorphism group
In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
of the graph. The colors remain labeled; it is the graph that is unlabeled.
There is an analogue of the
chromatic polynomial which counts the number of unlabeled colorings of a graph from a given finite color set.
If we interpret a coloring of a graph on vertices as a vector in , the action of an automorphism is a
permutation
In mathematics, a permutation of a set can mean one of two different things:
* an arrangement of its members in a sequence or linear order, or
* the act or process of changing the linear order of an ordered set.
An example of the first mean ...
of the coefficients in the coloring vector.
Properties
Upper bounds on the chromatic number
Assigning distinct colors to distinct vertices always yields a proper coloring, so
:
The only graphs that can be 1-colored are
edgeless graphs. A
complete graph
In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices i ...
of ''n'' vertices requires
colors. In an optimal coloring there must be at least one of the graph's ''m'' edges between every pair of color classes, so
:
More generally a family
of graphs is
-bounded if there is some function
such that the graphs
in
can be colored with at most
colors, where
is the
clique number of
. For the family of the perfect graphs this function is
.
The 2-colorable graphs are exactly the
bipartite graph
In the mathematics, mathematical field of graph theory, a bipartite graph (or bigraph) is a Graph (discrete mathematics), graph whose vertex (graph theory), vertices can be divided into two disjoint sets, disjoint and Independent set (graph theo ...
s, including
tree
In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, e.g., including only woody plants with secondary growth, only ...
s and forests.
By the four color theorem, every planar graph can be 4-colored.
A
greedy coloring
In the study of graph coloring problems in mathematics and computer science, a greedy coloring or sequential coloring is a coloring of the vertices of a graph formed by a greedy algorithm that considers the vertices of the graph in sequence an ...
shows that every graph can be colored with one more color than the maximum vertex
degree,
:
Complete graphs have
and
, and
odd cycles have
and
, so for these graphs this bound is best possible. In all other cases, the bound can be slightly improved;
Brooks' theorem
In graph theory, Brooks' theorem states a relationship between the maximum degree (graph theory), degree of a graph and its chromatic number. According to the theorem, in a connected graph in which every vertex has at most Δ neighbors, the vertic ...
states that
:
Brooks' theorem
In graph theory, Brooks' theorem states a relationship between the maximum degree (graph theory), degree of a graph and its chromatic number. According to the theorem, in a connected graph in which every vertex has at most Δ neighbors, the vertic ...
:
for a connected, simple graph ''G'', unless ''G'' is a complete graph or an odd cycle.
Lower bounds on the chromatic number
Several lower bounds for the chromatic bounds have been discovered over the years:
If ''G'' contains a
clique of size ''k'', then at least ''k'' colors are needed to color that clique; in other words, the chromatic number is at least the clique number:
:
For
perfect graph
In graph theory, a perfect graph is a Graph (discrete mathematics), graph in which the Graph coloring, chromatic number equals the size of the maximum clique, both in the graph itself and in every induced subgraph. In all graphs, the chromatic nu ...
s this bound is tight. Finding cliques is known as the
clique problem.
Hoffman's bound: Let
be a real symmetric matrix such that
whenever
is not an edge in
. Define
, where
are the largest and smallest eigenvalues of
. Define
, with
as above. Then:
:
: Let
be a positive semi-definite matrix such that
whenever
is an edge in
. Define
to be the least k for which such a matrix
exists. Then
:
Lovász number: The Lovász number of a complementary graph is also a lower bound on the chromatic number:
:
Fractional chromatic number: The fractional chromatic number of a graph is a lower bound on the chromatic number as well:
:
These bounds are ordered as follows:
:
Graphs with high chromatic number
Graphs with large
cliques
A clique ( AusE, CanE, or ; ), in the social sciences, is a small group of individuals who interact with one another and share similar interests rather than include others. Interacting with cliques is part of normative social development regardle ...
have a high chromatic number, but the opposite is not true. The
Grötzsch graph
In the mathematical field of graph theory, the Grötzsch graph is a triangle-free graph with 11 vertices, 20 edges, chromatic number 4, and crossing number 5. It is named after German mathematician Herbert Grötzsch, who used it as an example ...
is an example of a 4-chromatic graph without a triangle, and the example can be generalized to the
Mycielskians.
: Theorem (, , ): There exist triangle-free graphs with arbitrarily high chromatic number.
To prove this, both, Mycielski and Zykov, each gave a construction of an inductively defined family of
triangle-free graphs but with arbitrarily large chromatic number. constructed axis aligned boxes in
whose
intersection graph is triangle-free and requires arbitrarily many colors to be properly colored. This family of graphs is then called the Burling graphs. The same class of graphs is used for the construction of a family of triangle-free line segments in the plane, given by Pawlik et al. (2014). It shows that the chromatic number of its intersection graph is arbitrarily large as well. Hence, this implies that axis aligned boxes in
as well as line segments in
are not
''χ''-bounded.
From Brooks's theorem, graphs with high chromatic number must have high maximum degree. But colorability is not an entirely local phenomenon: A graph with high
girth
Girth may refer to:
Mathematics
* Girth (functional analysis), the length of the shortest centrally symmetric simple closed curve on the unit sphere of a Banach space
* Girth (geometry), the perimeter of a parallel projection of a shape
* Girth ...
looks locally like a tree, because all cycles are long, but its chromatic number need not be 2:
: Theorem (
Erdős): There exist graphs of arbitrarily high girth and chromatic number.
Bounds on the chromatic index
An edge coloring of ''G'' is a vertex coloring of its
line graph
In the mathematics, mathematical discipline of graph theory, the line graph of an undirected graph is another graph that represents the adjacencies between edge (graph theory), edges of . is constructed in the following way: for each edge i ...
, and vice versa. Thus,
:
There is a strong relationship between edge colorability and the graph's maximum degree
. Since all edges incident to the same vertex need their own color, we have
:
Moreover,
:
Kőnig's theorem:
if ''G'' is bipartite.
In general, the relationship is even stronger than what Brooks's theorem gives for vertex coloring:
:
Vizing's Theorem: A graph of maximal degree
has edge-chromatic number
or
.
Other properties
A graph has a ''k''-coloring if and only if it has an
acyclic orientation for which the
longest path has length at most ''k''; this is the
Gallai–Hasse–Roy–Vitaver theorem .
For planar graphs, vertex colorings are essentially dual to
nowhere-zero flows.
About infinite graphs, much less is known.
The following are two of the few results about infinite graph coloring:
*If all finite subgraphs of an
infinite graph ''G'' are ''k''-colorable, then so is ''G'', under the assumption of the
axiom of choice
In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, it is possible to construct a new set by choosing one element from e ...
. This is the
de Bruijn–Erdős theorem of .
*If a graph admits a full ''n''-coloring for every ''n'' ≥ ''n''
0, it admits an infinite full coloring .
Open problems
As stated above,
A conjecture of Reed from 1998 is that the value is essentially closer to the lower bound,
The
chromatic number of the plane, where two points are adjacent if they have unit distance, is unknown, although it is one of 5, 6, or 7. Other
open problems
In science and mathematics, an open problem or an open question is a known problem which can be accurately stated, and which is assumed to have an objective and verifiable solution, but which has not yet been solved (i.e., no solution for it is kno ...
concerning the chromatic number of graphs include the
Hadwiger conjecture stating that every graph with chromatic number ''k'' has a
complete graph
In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices i ...
on ''k'' vertices as a
minor, the
Erdős–Faber–Lovász conjecture bounding the chromatic number of unions of complete graphs that have at most one vertex in common to each pair, and the
Albertson conjecture that among ''k''-chromatic graphs the complete graphs are the ones with smallest
crossing number.
When Birkhoff and Lewis introduced the chromatic polynomial in their attack on the four-color theorem, they conjectured that for planar graphs ''G'', the polynomial
has no zeros in the region
perfect graph
In graph theory, a perfect graph is a Graph (discrete mathematics), graph in which the Graph coloring, chromatic number equals the size of the maximum clique, both in the graph itself and in every induced subgraph. In all graphs, the chromatic nu ...
s can be computed in
polynomial time
In theoretical computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations p ...
using
semidefinite programming.
Closed formulas for chromatic polynomials are known for many classes of graphs, such as forests, chordal graphs, cycles, wheels, and ladders, so these can be evaluated in polynomial time.
If the graph is planar and has low branch-width (or is nonplanar but with a known
branch-decomposition), then it can be solved in polynomial time using dynamic programming. In general, the time required is polynomial in the graph size, but exponential in the branch-width.
Exact algorithms
Brute-force search
In computer science, brute-force search or exhaustive search, also known as generate and test, is a very general problem-solving technique and algorithmic paradigm that consists of Iteration#Computing, systematically checking all possible candida ...
for a ''k''-coloring considers each of the
k^n assignments of ''k'' colors to ''n'' vertices and checks for each if it is legal. To compute the chromatic number and the chromatic polynomial, this procedure is used for every
k=1,\ldots,n-1, impractical for all but the smallest input graphs.
Using
dynamic programming and a bound on the number of
maximal independent sets, ''k''-colorability can be decided in time and space
O(2.4423^n). Using the principle of
inclusion–exclusion and
Yates's algorithm for the fast zeta transform, ''k''-colorability can be decided in time
O(2^n n) for any ''k''. Faster algorithms are known for 3- and 4-colorability, which can be decided in time
O(1.3289^n) and
O(1.7272^n), respectively. Exponentially faster algorithms are also known for 5- and 6-colorability, as well as for restricted families of graphs, including sparse graphs.
Contraction
The
contraction G/uv of a graph ''G'' is the graph obtained by identifying the vertices ''u'' and ''v'', and removing any edges between them. The remaining edges originally incident to ''u'' or ''v'' are now incident to their identification (''i.e.'', the new fused node ''uv''). This operation plays a major role in the analysis of graph coloring.
The chromatic number satisfies the
recurrence relation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
:
:
\chi(G) = \text \
due to , where ''u'' and ''v'' are non-adjacent vertices, and
G+uv is the graph with the edge added. Several algorithms are based on evaluating this recurrence and the resulting computation tree is sometimes called a Zykov tree. The running time is based on a heuristic for choosing the vertices ''u'' and ''v''.
The chromatic polynomial satisfies the following recurrence relation
:
P(G-uv, k)= P(G/uv, k)+ P(G, k)
where ''u'' and ''v'' are adjacent vertices, and
G-uv is the graph with the edge removed.
P(G - uv, k) represents the number of possible proper colorings of the graph, where the vertices may have the same or different colors. Then the proper colorings arise from two different graphs. To explain, if the vertices ''u'' and ''v'' have different colors, then we might as well consider a graph where ''u'' and ''v'' are adjacent. If ''u'' and ''v'' have the same colors, we might as well consider a graph where ''u'' and ''v'' are contracted. Tutte's curiosity about which other graph properties satisfied this recurrence led him to discover a bivariate generalization of the chromatic polynomial, the
Tutte polynomial.
These expressions give rise to a recursive procedure called the ''deletion–contraction algorithm'', which forms the basis of many algorithms for graph coloring. The running time satisfies the same recurrence relation as the
Fibonacci numbers
In mathematics, the Fibonacci sequence is a sequence in which each element is the sum of the two elements that precede it. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted . Many writers begin the s ...
, so in the worst case the algorithm runs in time within a polynomial factor of
\left(\tfrac2\right)^=O(1.6180^) for ''n'' vertices and ''m'' edges. The analysis can be improved to within a polynomial factor of the number
t(G) of
spanning trees of the input graph. In practice,
branch and bound
Branch and bound (BB, B&B, or BnB) is a method for solving optimization problems by breaking them down into smaller sub-problems and using a bounding function to eliminate sub-problems that cannot contain the optimal solution.
It is an algorithm ...
strategies and
graph isomorphism
In graph theory, an isomorphism of graphs ''G'' and ''H'' is a bijection between the vertex sets of ''G'' and ''H''
: f \colon V(G) \to V(H)
such that any two vertices ''u'' and ''v'' of ''G'' are adjacent in ''G'' if and only if f(u) and f(v) a ...
rejection are employed to avoid some recursive calls. The running time depends on the heuristic used to pick the vertex pair.
Greedy coloring

The
greedy algorithm
A greedy algorithm is any algorithm that follows the problem-solving heuristic of making the locally optimal choice at each stage. In many problems, a greedy strategy does not produce an optimal solution, but a greedy heuristic can yield locally ...
considers the vertices in a specific order
v_1, ...,
v_n and assigns to
v_i the smallest available color not used by
v_i's neighbours among
v_1, ...,
v_, 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
\chi(G) colors. On the other hand, greedy colorings can be arbitrarily bad; for example, the
crown graph on ''n'' vertices can be 2-colored, but has an ordering that leads to a greedy coloring with
n/2 colors.
For
chordal graphs, and for special cases of chordal graphs such as
interval graphs and
indifference graphs, the greedy coloring algorithm can be used to find optimal colorings in polynomial time, by choosing the vertex ordering to be the reverse of a
perfect elimination ordering for the graph. The
perfectly orderable graphs generalize this property, but it is NP-hard to find a perfect ordering of these graphs.
If the vertices are ordered according to their
degrees, the resulting greedy coloring uses at most
\text_i \text
\ colors, at most one more than the graph's maximum degree. This heuristic is sometimes called the Welsh–Powell algorithm. Another heuristic due to
Brélaz establishes the ordering dynamically while the algorithm proceeds, choosing next the vertex adjacent to the largest number of different colors. 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.
The maximum (worst) number of colors that can be obtained by the greedy algorithm, by using a vertex ordering chosen to maximize this number, is called the
Grundy number of a graph.
Heuristic algorithms
Two well-known polynomial-time heuristics for graph colouring are the
DSatur and
recursive largest first (RLF) algorithms.
Similarly to the
greedy colouring algorithm, DSatur colours the
vertices of a
graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discret ...
one after another, expending a previously unused colour when needed. Once a new
vertex has been coloured, the algorithm determines which of the remaining uncoloured vertices has the highest number of different colours in its neighbourhood and colours this vertex next. This is defined as the ''degree of saturation'' of a given vertex.
The
recursive largest first algorithm operates in a different fashion by constructing each color class one at a time. It does this by identifying a
maximal independent set of vertices in the graph using specialised heuristic rules. It then assigns these vertices to the same color and removes them from the graph. These actions are repeated on the remaining subgraph until no vertices remain.
The worst-case complexity of DSatur is
O(n^2), where
n is the number of vertices in the graph. The algorithm can also be implemented using a binary heap to store saturation degrees, operating in
O((n+m)\log n) where
m is the number of edges in the graph. This produces much faster runs with sparse graphs. The overall complexity of RLF is slightly higher than
DSatur at
O(mn).
DSatur and RLF are
exact for
bipartite,
cycle, and
wheel graphs.
Parallel and distributed algorithms
It is known that a -chromatic graph can be -colored in the deterministic LOCAL model, in
O(n^). rounds, with
\alpha = \left\lfloor \frac \right\rfloor. A matching lower bound of
\Omega(n^) rounds is also known. This lower bound holds even if quantum computers that can exchange quantum information, possibly with a pre-shared entangled state, are allowed.
In the field of
distributed algorithm A distributed algorithm is an algorithm designed to run on computer hardware constructed from interconnected processors. Distributed algorithms are used in different application areas of distributed computing, such as telecommunications, scientifi ...
s, graph coloring is closely related to the problem of
symmetry breaking
In physics, symmetry breaking is a phenomenon where a disordered but Symmetry in quantum mechanics, symmetric state collapses into an ordered, but less symmetric state. This collapse is often one of many possible Bifurcation theory, bifurcatio ...
. The current state-of-the-art randomized algorithms are faster for sufficiently large maximum degree Δ than deterministic algorithms. The fastest randomized algorithms employ the
multi-trials technique by Schneider and Wattenhofer.
In a
symmetric graph, a
deterministic
Determinism is the metaphysical view that all events within the universe (or multiverse) can occur only in one possible way. Deterministic theories throughout the history of philosophy have developed from diverse and sometimes overlapping mo ...
distributed algorithm cannot find a proper vertex coloring. Some auxiliary information is needed in order to break symmetry. A standard assumption is that initially each node has a ''unique identifier'', for example, from the set . Put otherwise, we assume that we are given an ''n''-coloring. The challenge is to ''reduce'' the number of colors from ''n'' to, e.g., Δ + 1. The more colors are employed, e.g. ''O''(Δ) instead of Δ + 1, the fewer communication rounds are required.
A straightforward distributed version of the greedy algorithm for (Δ + 1)-coloring requires Θ(''n'') communication rounds in the worst case – information may need to be propagated from one side of the network to another side.
The simplest interesting case is an ''n''-
cycle. Richard Cole and
Uzi Vishkin
Uzi Vishkin (; born 1953) is a computer scientist at the University of Maryland, College Park, where he is Professor of Electrical and Computer Engineering at the University of Maryland Institute for Advanced Computer Studies (UMIACS). Uzi Vishkin ...
show that there is a distributed algorithm that reduces the number of colors from ''n'' to ''O''(log ''n'') in one synchronous communication step. By iterating the same procedure, it is possible to obtain a 3-coloring of an ''n''-cycle in ''O''( ''n'') communication steps (assuming that we have unique node identifiers).
The function ,
iterated logarithm, is an extremely slowly growing function, "almost constant". Hence the result by Cole and Vishkin raised the question of whether there is a ''constant-time'' distributed algorithm for 3-coloring an ''n''-cycle. showed that this is not possible: any deterministic distributed algorithm requires Ω( ''n'') communication steps to reduce an ''n''-coloring to a 3-coloring in an ''n''-cycle.
The technique by Cole and Vishkin can be applied in arbitrary bounded-degree graphs as well; the running time is poly(Δ) + ''O''( ''n''). The technique was extended to
unit disk graphs by Schneider and Wattenhofer. The fastest deterministic algorithms for (Δ + 1)-coloring for small Δ are due to Leonid Barenboim, Michael Elkin and Fabian Kuhn. The algorithm by Barenboim et al. runs in time ''O''(Δ) + (''n'')/2, which is optimal in terms of ''n'' since the constant factor 1/2 cannot be improved due to Linial's lower bound. use network decompositions to compute a Δ+1 coloring in time
2
^ .
The problem of edge coloring has also been studied in the distributed model. achieve a (2Δ − 1)-coloring in ''O''(Δ + ''n'') time in this model. The lower bound for distributed vertex coloring due to applies to the distributed edge coloring problem as well.
Decentralized algorithms
Decentralized algorithms are ones where no
message passing
In computer science, message passing is a technique for invoking behavior (i.e., running a program) on a computer. The invoking program sends a message to a process (which may be an actor or object) and relies on that process and its supporting ...
is allowed (in contrast to distributed algorithms where local message passing takes places), and efficient decentralized algorithms exist that will color a graph if a proper coloring exists. These assume that a vertex is able to sense whether any of its neighbors are using the same color as the vertex i.e., whether a local conflict exists. This is a mild assumption in many applications e.g. in wireless channel allocation it is usually reasonable to assume that a station will be able to detect whether other interfering transmitters are using the same channel (e.g. by measuring the SINR). This sensing information is sufficient to allow algorithms based on learning automata to find a proper graph coloring with probability one.
Computational complexity
Graph coloring is computationally hard. It is
NP-complete
In computational complexity theory, NP-complete problems are the hardest of the problems to which ''solutions'' can be verified ''quickly''.
Somewhat more precisely, a problem is NP-complete when:
# It is a decision problem, meaning that for any ...
to decide if a given graph admits a ''k''-coloring for a given ''k'' except for the cases ''k'' ∈ . In particular, it is NP-hard to compute the chromatic number.
[; .] The 3-coloring problem remains NP-complete even on 4-regular
planar graph
In graph theory, a planar graph is a graph (discrete mathematics), graph that can be graph embedding, embedded in the plane (geometry), plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. ...
s. On graphs with maximal degree 3 or less, however,
Brooks' theorem
In graph theory, Brooks' theorem states a relationship between the maximum degree (graph theory), degree of a graph and its chromatic number. According to the theorem, in a connected graph in which every vertex has at most Δ neighbors, the vertic ...
implies that the 3-coloring problem can be solved in linear time. Further, for every ''k'' > 3, a ''k''-coloring of a planar graph exists by the
four color theorem
In mathematics, the four color theorem, or the four color map theorem, states that no more than four colors are required to color the regions of any map so that no two adjacent regions have the same color. ''Adjacent'' means that two regions shar ...
, and it is possible to find such a coloring in polynomial time. However, finding the
lexicographically smallest 4-coloring of a planar graph is NP-complete.
The best known
approximation algorithm
In computer science and operations research, approximation algorithms are efficient algorithms that find approximate solutions to optimization problems (in particular NP-hard problems) with provable guarantees on the distance of the returned sol ...
computes a coloring of size at most within a factor ''O''(''n''(log log ''n'')
2(log n)
−3) of the chromatic number. For all ''ε'' > 0, approximating the chromatic number within ''n''
1−''ε'' is
NP-hard
In computational complexity theory, a computational problem ''H'' is called NP-hard if, for every problem ''L'' which can be solved in non-deterministic polynomial-time, there is a polynomial-time reduction from ''L'' to ''H''. That is, assumi ...
.
It is also NP-hard to color a 3-colorable graph with 5 colors, 4-colorable graph with 7 colours, and a ''k''-colorable graph with
\textstyle\binom k - 1 colors for ''k'' ≥ 5.
Computing the coefficients of the chromatic polynomial is
♯P-hard. In fact, even computing the value of
\chi(G,k) is ♯P-hard at any
rational point
In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the fiel ...
''k'' except for ''k'' = 1 and ''k'' = 2. There is no
FPRAS for evaluating the chromatic polynomial at any rational point ''k'' ≥ 1.5 except for ''k'' = 2 unless
NP =
RP.
For edge coloring, the proof of Vizing's result gives an algorithm that uses at most Δ+1 colors. However, deciding between the two candidate values for the edge chromatic number is NP-complete. In terms of approximation algorithms, Vizing's algorithm shows that the edge chromatic number can be approximated to within 4/3,
and the hardness result shows that no (4/3 − ''ε'')-algorithm exists for any ''ε > 0'' unless
P = NP. These are among the oldest results in the literature of approximation algorithms, even though neither paper makes explicit use of that notion.
Applications
Scheduling
Vertex coloring models to a number of
scheduling problems. In the cleanest form, a given set of jobs need to be assigned to time slots, each job requires one such slot. Jobs can be scheduled in any order, but pairs of jobs may be in ''conflict'' in the sense that they may not be assigned to the same time slot, for example because they both rely on a shared resource. The corresponding graph contains a vertex for every job and an edge for every conflicting pair of jobs. The chromatic number of the graph is exactly the minimum ''makespan'', the optimal time to finish all jobs without conflicts.
Details of the scheduling problem define the structure of the graph. For example, when assigning aircraft to flights, the resulting conflict graph is an
interval graph, so the coloring problem can be solved efficiently. In
bandwidth allocation
Bandwidth allocation is the process of assigning radio frequencies to different applications. The radio spectrum is a finite resource, which means there is great need for an effective allocation process. In the United States, the Federal Commun ...
to radio stations, the resulting conflict graph is a
unit disk graph, so the coloring problem is 3-approximable.
Register allocation
A
compiler
In computing, a compiler is a computer program that Translator (computing), translates computer code written in one programming language (the ''source'' language) into another language (the ''target'' language). The name "compiler" is primaril ...
is a
computer program
A computer program is a sequence or set of instructions in a programming language for a computer to Execution (computing), execute. It is one component of software, which also includes software documentation, documentation and other intangibl ...
that translates one
computer language
A computer language is a formal language used to communicate with a computer. Types of computer languages include:
* Software construction#Construction languages, Construction language – all forms of communication by which a human can Comput ...
into another. To improve the execution time of the resulting code, one of the techniques of
compiler optimization
An optimizing compiler is a compiler designed to generate code that is optimized in aspects such as minimizing program execution time, memory usage, storage size, and power consumption. Optimization is generally implemented as a sequence of op ...
is
register allocation
In compiler optimization, register allocation is the process of assigning local automatic variables and Expression (computer science), expression results to a limited number of processor registers.
Register allocation can happen over a basic bloc ...
, where the most frequently used values of the compiled program are kept in the fast
processor register
A processor register is a quickly accessible location available to a computer's processor. Registers usually consist of a small amount of fast storage, although some registers have specific hardware functions, and may be read-only or write-onl ...
s. Ideally, values are assigned to registers so that they can all reside in the registers when they are used.
The textbook approach to this problem is to model it as a graph coloring problem. The compiler constructs an ''interference graph'', where vertices are variables and an edge connects two vertices if they are needed at the same time. If the graph can be colored with ''k'' colors then any set of variables needed at the same time can be stored in at most ''k'' registers.
Other applications
The problem of coloring a graph arises in many practical areas such as sports scheduling, designing seating plans, exam timetabling, the scheduling of taxis, and solving
Sudoku
Sudoku (; ; originally called Number Place) is a logic puzzle, logic-based, combinatorics, combinatorial number-placement puzzle. In classic Sudoku, the objective is to fill a 9 × 9 grid with digits so that each column, each row, and ...
puzzles.
Other colorings
Ramsey theory
An important class of ''improper'' coloring problems is studied in
Ramsey theory
Ramsey theory, named after the British mathematician and philosopher Frank P. Ramsey, is a branch of the mathematical field of combinatorics that focuses on the appearance of order in a substructure given a structure of a known size. Problems in R ...
, where the graph's edges are assigned to colors, and there is no restriction on the colors of incident edges. A simple example is the
theorem on friends and strangers, which states that in any coloring of the edges of
K_6, the complete graph of six vertices, there will be a monochromatic triangle; often illustrated by saying that any group of six people either has three mutual strangers or three mutual acquaintances. Ramsey theory is concerned with generalisations of this idea to seek regularity amid disorder, finding general conditions for the existence of monochromatic subgraphs with given structure.
Modular Coloring
Modular coloring is a type of graph coloring in which the color of each vertex is the sum of the colors of its adjacent vertices.
Let be a number of colors where
\mathbb_k is the set of integers modulo k consisting of the elements (or colors) . First, we color each vertex in G using the elements of
\mathbb_k, allowing two adjacent vertices to be assigned the same color. In other words, we want c to be a coloring such that c: V(G) →
\mathbb_k where adjacent vertices can be assigned the same color.
For each vertex v in G, the color sum of , is the sum of all of the adjacent vertices to v mod k. The color sum of v is denoted by
:
where u is an arbitrary vertex in the neighborhood of v, N(v). We then color each vertex with the new coloring determined by the sum of the adjacent vertices. The graph G has a modular k-coloring if, for every pair of adjacent vertices a,b, σ(a) ≠ σ(b). The modular chromatic number of G, mc(G), is the minimum value of k such that there exists a modular k-coloring of G.<
For example, let there be a vertex v adjacent to vertices with the assigned colors 0, 1, 1, and 3 mod 4 (k=4). The color sum would be σ(v) = 0 + 1 + 1+ 3 mod 4 = 5 mod 4 = 1. This would be the new color of vertex v. We would repeat this process for every vertex in G. If two adjacent vertices have equal color sums, G does not have a modulo 4 coloring. If none of the adjacent vertices have equal color sums, G has a modulo 4 coloring.
Other colorings
;
Adjacent-vertex-distinguishing-total coloring : A total coloring with the additional restriction that any two adjacent vertices have different color sets
;
Acyclic coloring : Every 2-chromatic subgraph is acyclic
;
B-coloring : a coloring of the vertices where each color class contains a vertex that has a neighbor in all other color classes.
; Bounded Coloring : a coloring in which the number of vertices per color is bounded
;
Circular coloring : Motivated by task systems in which production proceeds in a cyclic way
;
Cocoloring : An improper vertex coloring where every color class induces an independent set or a clique
;
Complete coloring : Every pair of colors appears on at least one edge
;
Defective coloring : An improper vertex coloring where every color class induces a bounded degree subgraph.
;
Distinguishing coloring : An improper vertex coloring that destroys all the symmetries of the graph
;
Equitable coloring : The sizes of color classes differ by at most one
;
Exact coloring : Every pair of colors appears on exactly one edge
;
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
;
Hamiltonian coloring : Uses the length of the longest path between two vertices, also known as the detour distance
;
Harmonious coloring : Every pair of colors appears on at most one edge
;
Incidence coloring: Each adjacent incidence of vertex and edge is colored with distinct colors
;
Inherited vertex coloring : A set of vertex colorings induced by perfect matchings of
edge-colored Graphs.
;
Interval edge coloring : A color of edges meeting in a common vertex must be contiguous
;
List coloring: Each vertex chooses from a list of colors
;
List edge-coloring:Each edge chooses from a list of colors
;
L(''h'', ''k'')-coloring: Difference of colors at adjacent vertices is at least ''h'' and difference of colors of vertices at a distance two is at least ''k''. A particular case is
L(2,1)-coloring.
;
Oriented coloring : Takes into account orientation of edges of the graph
;
Path coloring : Models a routing problem in graphs
;
Radio coloring : Sum of the distance between the vertices and the difference of their colors is greater than ''k'' + 1, where ''k'' is a positive integer.
;
Rank coloring : If two vertices have the same color ''i'', then every path between them contain a vertex with color greater than ''i''
;
Subcoloring : An improper vertex coloring where every color class induces a union of cliques
;
Sum coloring : The criterion of minimalization is the sum of colors
;
Star coloring : Every 2-chromatic subgraph is a disjoint collection of
stars
A star is a luminous spheroid of plasma held together by self-gravity. The nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night; their immense distances from Earth make them appear as fixed points of ...
;
Strong coloring : Every color appears in every partition of equal size exactly once
;
Strong edge coloring : Edges are colored such that each color class induces a matching (equivalent to coloring the square of the line graph)
;
''T''-coloring : Absolute value of the difference between two colors of adjacent vertices must not belong to fixed set ''T''
;
Total coloring :Vertices and edges are colored
;
Centered coloring: Every connected
induced subgraph
In graph theory, an induced subgraph of a graph is another graph, formed from a subset of the vertices of the graph and ''all'' of the edges, from the original graph, connecting pairs of vertices in that subset.
Definition
Formally, let G=(V,E) ...
has a color that is used exactly once
;
Triangle-free edge coloring: The edges are colored so that each color class forms a
triangle-free subgraph
;
Weak coloring : An improper vertex coloring where every non-isolated node has at least one neighbor with a different color
Coloring can also be considered for
signed graph
In the area of graph theory in mathematics, a signed graph is a graph in which each edge has a positive or negative sign.
A signed graph is balanced if the product of edge signs around every cycle is positive. The name "signed graph" and the no ...
s and
gain graphs.
See also
*
Critical graph
*
Graph coloring game
*
Graph homomorphism
*
Hajós construction
*
Mathematics of Sudoku
Mathematics can be used to study Sudoku puzzles to answer questions such as "How many filled Sudoku grids are there?", "What is the minimal number of clues in a valid puzzle?" and "In what ways can Sudoku grids be symmetric?" through the use o ...
*
Multipartite graph
*
Uniquely colorable graph
Notes
References
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*
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* (= ''Indag. Math.'' 13)
*
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*
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*
* . Translated into English in ''Amer. Math. Soc. Translation'', 1952, .
External links
''GCol''An open-source python library for graph coloring.
''High-Performance Graph Colouring Algorithms''Suite of 8 different algorithms (implemented in C++) used in the book
A Guide to Graph Colouring: Algorithms and Applications' (Springer International Publishers, 2015).
''CoLoRaTiOn''by Jim Andrews and Mike Fellows is a graph coloring puzzle
Code for efficiently computing Tutte, Chromatic and Flow Polynomials by Gary Haggard, David J. Pearce and Gordon Royle
A graph coloring Web Appby Jose Antonio Martin H.
{{DEFAULTSORT:Graph Coloring
Coloring
NP-complete problems
NP-hard problems
Computational problems in graph theory
Extensions and generalizations of graphs