In
graph theory
In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...
, graph coloring is a special case of
graph labeling; it is an 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 discre ...
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 are of 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 are of the same color, and a face coloring of a
planar graph
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 ...
assigns a color to each face 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, and a face coloring of a plane graph is just a vertex coloring of its
dual
Dual or Duals may refer to:
Paired/two things
* Dual (mathematics), a notion of paired concepts that mirror one another
** Dual (category theory), a formalization of mathematical duality
*** see more cases in :Duality theories
* Dual (grammatical ...
. However, non-vertex coloring problems are often stated and studied as-is. This is partly
pedagogical, 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 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 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. ...
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
Glossary of graph theory.''
History
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
Francis Guthrie (born 22 January 1831 in London; d. 19 October 1899 in Claremont, Cape Town) was a South African mathematician and botanist who first posed the Four Colour Problem in 1852. He studied mathematics under Augustus De Morgan, and ...
postulated the
four color conjecture
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 sh ...
, 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. The same year,
Alfred Kempe 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, r ...
and later President of the London Mathematical Society.
In 1890,
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 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 also noteworthy for being the first major computer-aided proof.
In 1912,
George David 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, 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
Seymour may refer to:
Places Australia
*Seymour, Victoria, a township
*Electoral district of Seymour, a former electoral district in Victoria
*Rural City of Seymour, a former local government area in Victoria
*Seymour, Tasmania, a locality
...
, and
Thomas in 2002.
Graph coloring has been studied as an algorithmic problem since the early 1970s: the chromatic number problem (see
below
Below may refer to:
*Earth
* Ground (disambiguation)
* Soil
* Floor
* Bottom (disambiguation)
* Less than
*Temperatures below freezing
* Hell or underworld
People with the surname
* Ernst von Below (1863–1955), German World War I general
* Fr ...
) 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 compilers, was introduced in 1981.
Definition and terminology
Vertex coloring
When used without any qualification, a coloring of a graph is almost always 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
Loop or LOOP may refer to:
Brands and enterprises
* Loop (mobile), a Bulgarian virtual network operator and co-founder of Loop Live
* Loop, clothing, a company founded by Carlos Vasquez in the 1990s and worn by Digable Planets
* Loop Mobile, ...
(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 (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
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 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 + 4â‹…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 an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...
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 sh ...
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 .
Unlabeled coloring
An unlabeled coloring of a graph is an
orbit
In celestial mechanics, an orbit 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 object or position in space such a ...
of a coloring under the action of the
automorphism group of the graph. Note that 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 is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or p ...
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 graph
In the mathematical field of graph theory, the term "null graph" may refer either to the order- zero graph, or alternatively, to any edgeless graph (the latter is sometimes called an "empty graph").
Order-zero graph
The order-zero graph, , is ...
s. A
complete graph 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
:
If ''G'' contains a
clique
A clique ( AusE, CanE, or ), in the social sciences, is a group of individuals who interact with one another and share similar interests. Interacting with cliques is part of normative social development regardless of gender, ethnicity, or popula ...
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 in which the chromatic number of every induced subgraph equals the order of the largest clique of that subgraph ( clique number). Equivalently stated in symbolic terms an arbitrary graph G=(V,E) is per ...
s this bound is tight. Finding cliques is known as the
clique problem.
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, for the family of the perfect graphs this function is
.
The 2-colorable graphs are exactly the
bipartite graphs, 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, including only woody plants with secondary growth, plants that are ...
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 a ...
shows that every graph can be colored with one more color than the maximum vertex
degree
Degree may refer to:
As a unit of measurement
* Degree (angle), a unit of angle measurement
** Degree of geographical latitude
** Degree of geographical longitude
* Degree symbol (°), a notation used in science, engineering, and mathemati ...
,
:
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 states that
:
Brooks' theorem:
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:
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 have a high chromatic number, but the opposite is not true. The
Grötzsch graph is an example of a 4-chromatic graph without a triangle, and the example can be generalized to the
Mycielskian
In the mathematical area of graph theory, the Mycielskian or Mycielski graph of an undirected graph is a larger graph formed from it by a construction of . The construction preserves the property of being triangle-free but increases the chroma ...
s.
: 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. Burling (1965) 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 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 , 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,
:
KÅ‘nig'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, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
. 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 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, the
Erdős–Faber–Lovász conjecture
In graph theory, the Erdős–Faber–Lovász conjecture is a problem about graph coloring, named after Paul Erdős, Vance Faber, and László Lovász, who formulated it in 1972.. It says:
:If complete graphs, each having exactly vertices, have ...
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