HOME

TheInfoList



OR:

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 In the mathematical discipline of graph theory, a graph labelling is the assignment of labels, traditionally represented by integers, to edges and/or vertices of a graph. Formally, given a graph , a vertex labelling is a function of to a set o ...
; 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 In graph theory, an edge coloring of a graph is an assignment of "colors" to the edges of the graph so that no two incident edges have the same color. For example, the figure to the right shows an edge coloring of a graph by the colors red, blu ...
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 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 In the mathematical discipline of graph theory, the line graph of an undirected graph is another graph that represents the adjacencies between edges of . is constructed in the following way: for each edge in , make a vertex in ; for every ...
, 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 as ...
, 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 A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes. Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Although ...
, 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. T ...
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 (; ja, 数独, sūdoku, digit-single; originally called Number Place) is a logic-based, combinatorial number-placement puzzle. In classic Sudoku, the objective is to fill a 9 × 9 grid with digits so that each column, each row ...
. 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, 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 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 ...
, 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 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 Edward K ...
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,
Heawood Heawood is a surname. Notable people with the surname include: *Jonathan Heawood, British journalist *Percy John Heawood (1861–1955), British mathematician **Heawood conjecture **Heawood graph **Heawood number In mathematics, the Heawood number ...
pointed out that Kempe's argument was wrong. However, in that paper he proved the
five color theorem The five color theorem is a result from graph theory that given a plane separated into regions, such as a political map of the countries of the world, the regions may be colored using no more than five colors in such a way that no two adjacent reg ...
, 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 Kenneth Ira Appel (October 8, 1932 – April 19, 2013) was an American mathematician who in 1976, with colleague Wolfgang Haken at the University of Illinois at Urbana–Champaign, solved one of the most famous problems in mathematics, the fou ...
and
Wolfgang Haken Wolfgang Haken (June 21, 1928 – October 2, 2022) was a German American mathematician who specialized in topology, in particular 3-manifolds. Biography Haken was born in Berlin, Germany. His father was Werner Haken, a physicist who had Max ...
. 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 George David Birkhoff (March 21, 1884 – November 12, 1944) was an American mathematician best known for what is now called the ergodic theorem. Birkhoff was one of the most important leaders in American mathematics in his generation, and during ...
introduced the
chromatic polynomial The chromatic polynomial is a graph polynomial studied in algebraic graph theory, a branch of mathematics. It counts the number of graph colorings as a function of the number of colors and was originally defined by George David Birkhoff to s ...
to study the coloring problems, which was generalised to the
Tutte polynomial The Tutte polynomial, also called the dichromate or the Tutte–Whitney polynomial, is a graph polynomial. It is a polynomial in two variables which plays an important role in graph theory. It is defined for every undirected graph G and contai ...
by
Tutte William Thomas Tutte OC FRS FRSC (; 14 May 1917 – 2 May 2002) was an English and Canadian codebreaker and mathematician. During the Second World War, he made a brilliant and fundamental advance in cryptanalysis of the Lorenz cipher, a majo ...
, important structures in
algebraic graph theory Algebraic graph theory is a branch of mathematics in which algebraic methods are applied to problems about graphs. This is in contrast to geometric, combinatoric, or algorithmic approaches. There are three main branches of algebraic graph th ...
. 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 Claude Jacques Berge (5 June 1926 – 30 June 2002) was a French mathematician, recognized as one of the modern founders of combinatorics and graph theory. Biography and professional history Claude Berge's parents were André Berge and Geneviève ...
formulated another conjecture about graph coloring, the ''strong perfect graph conjecture'', originally motivated by an
information-theoretic Information theory is the scientific study of the quantification, storage, and communication of information. The field was originally established by the works of Harry Nyquist and Ralph Hartley, in the 1920s, and Claude Shannon in the 1940s. T ...
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 Robertson may refer to: People * Robertson (surname) (includes a list of people with this name) * Robertson (given name) * Clan Robertson, a Scottish clan * Robertson, stage name of Belgian magician Étienne-Gaspard Robert (1763–1837) Places ...
, 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 below) is one of
Karp's 21 NP-complete problems In computational complexity theory, Karp's 21 NP-complete problems are a set of computational problems which are NP-complete. In his 1972 paper, "Reducibility Among Combinatorial Problems", Richard Karp used Stephen Cook's 1971 theorem that the b ...
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 results to a limited number of processor registers. Register allocation can happen over a basic block (''local register allocatio ...
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 Edge or EDGE may refer to: Technology Computing * Edge computing, a network load-balancing system * Edge device, an entry point to a computer network * Adobe Edge, a graphical development application * Microsoft Edge, a web browser developed ...
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, an ...
(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 languag ...
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 + 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 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 example ...
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 In the mathematical field of graph theory, a cubic graph is a graph in which all vertices have degree three. In other words, a cubic graph is a 3-regular graph. Cubic graphs are also called trivalent graphs. A bicubic graph is a cubic bi ...
. The four color theorem 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 as ...
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. Note that the colors remain labeled; it is the graph that is unlabeled. There is an analogue of the
chromatic polynomial The chromatic polynomial is a graph polynomial studied in algebraic graph theory, a branch of mathematics. It counts the number of graph colorings as a function of the number of colors and was originally defined by George David Birkhoff to s ...
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 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 : 1 \le \chi(G) \le n. 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 th ...
s. 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 is ...
K_n of ''n'' vertices requires \chi(K_n)=n colors. In an optimal coloring there must be at least one of the graph's ''m'' edges between every pair of color classes, so : \chi(G)(\chi(G)-1) \le 2m. 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 popular ...
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: : \chi(G) \ge \omega(G). 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 perfe ...
s this bound is tight. Finding cliques is known as the
clique problem In computer science, the clique problem is the computational problem of finding cliques (subsets of vertices, all adjacent to each other, also called complete subgraphs) in a graph. It has several different formulations depending on which cli ...
. More generally a family \mathcal of graphs is \chi-bounded if there is some function c such that the graphs G in \mathcal can be colored with at most c(\omega(G)) colors, for the family of the perfect graphs this function is c(\omega(G))=\omega(G). 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 an ...
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 mathematics ...
, : \chi(G) \le \Delta(G) + 1. Complete graphs have \chi(G)=n and \Delta(G)=n-1, and
odd cycle In graph theory, a cycle graph or circular graph is a graph that consists of a single cycle, or in other words, some number of vertices (at least 3, if the graph is simple) connected in a closed chain. The cycle graph with vertices is called ...
s have \chi(G)=3 and \Delta(G)=2, 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 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 vertices can be colored with ...
states that :
Brooks' theorem In graph theory, Brooks' theorem states a relationship between the maximum 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 vertices can be colored with ...
: \chi (G) \le \Delta (G) 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 W be a real symmetric matrix such that W_ = 0 whenever (i,j) is not an edge in G. Define \chi_W(G) = 1 - \tfrac, where \lambda_(W), \lambda_(W) are the largest and smallest eigenvalues of W. Define \chi_H(G) = \max_W \chi_W(G), with W as above. Then: : \chi_H(G)\leq \chi(G). : Let W be a positive semi-definite matrix such that W_ \le -\tfrac whenever (i,j) is an edge in G. Define \chi_V(G) to be the least k for which such a matrix W exists. Then : \chi_V(G)\leq \chi(G).
Lovász number In graph theory, the Lovász number of a graph is a real number that is an upper bound on the Shannon capacity of the graph. It is also known as Lovász theta function and is commonly denoted by \vartheta(G), using a script form of the Greek letter ...
: The Lovász number of a complementary graph is also a lower bound on the chromatic number: : \vartheta(\bar) \leq \chi(G).
Fractional chromatic number Fractional coloring is a topic in a young branch of graph theory known as fractional graph theory. It is a generalization of ordinary graph coloring. In a traditional graph coloring, each vertex in a graph is assigned some color, and adjacent v ...
: The fractional chromatic number of a graph is a lower bound on the chromatic number as well: : \chi_f(G) \leq \chi(G). These bounds are ordered as follows: : \chi_H(G) \leq \chi_V(G) \leq \vartheta(\bar) \leq \chi_f(G) \leq \chi(G).


Graphs with high chromatic number

Graphs with large
cliques 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 popular ...
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
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 chromatic ...
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 graph In the mathematical area of graph theory, a triangle-free graph is an undirected graph in which no three vertices form a triangle of edges. Triangle-free graphs may be equivalently defined as graphs with clique number ≤ 2, graphs with g ...
s but with arbitrarily large chromatic number. Burling (1965) constructed axis aligned boxes in \mathbb^ whose
intersection graph In graph theory, an intersection graph is a graph that represents the pattern of intersections of a family of sets. Any graph can be represented as an intersection graph, but some important special classes of graphs can be defined by the types o ...
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 \mathbb^ as well as line segments in \mathbb^ are not
χ-bounded In graph theory, a \chi-bounded family \mathcal of graphs is one for which there is some function c such that, for every integer t the graphs in \mathcal with t=\omega(G) (clique number) can be colored with at most c(t) colors. This concept and its ...
. 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 In the mathematical discipline of graph theory, the line graph of an undirected graph is another graph that represents the adjacencies between edges of . is constructed in the following way: for each edge in , make a vertex in ; for every ...
L(G), and vice versa. Thus, :\chi'(G)=\chi(L(G)). There is a strong relationship between edge colorability and the graph's maximum degree \Delta(G). Since all edges incident to the same vertex need their own color, we have : \chi'(G) \ge \Delta(G). Moreover, : Kőnig's theorem: \chi'(G) = \Delta(G) 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 \Delta has edge-chromatic number \Delta or \Delta+1.


Other properties

A graph has a ''k''-coloring if and only if it has an
acyclic orientation In graph theory, an acyclic orientation of an undirected graph is an assignment of a direction to each edge (an orientation) that does not form any directed cycle and therefore makes it into a directed acyclic graph. Every graph has an acyclic orie ...
for which the
longest path In graph theory and theoretical computer science, the longest path problem is the problem of finding a simple path of maximum length in a given graph. A path is called ''simple'' if it does not have any repeated vertices; the length of a path ma ...
has length at most ''k''; this is the
Gallai–Hasse–Roy–Vitaver theorem In graph theory, the Gallai–Hasse–Roy–Vitaver theorem is a form of duality between the colorings of the vertices of a given undirected graph and the orientations of its edges. It states that the minimum number of colors needed to properly c ...
. 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 This is a glossary of graph theory. Graph theory is the study of graphs, systems of nodes or vertices connected in pairs by lines or edges. Symbols A B ...
''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, \omega(G) \le \chi(G) \le \Delta(G) + 1. A conjecture of Reed from 1998 is that the value is essentially closer to the lower bound, \chi(G) \le \left\lceil \frac \right\rceil. 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 There are several conjectures known as the Hadwiger conjecture or Hadwiger's conjecture. They include: * Hadwiger conjecture (graph theory), a relationship between the number of colors needed by a given graph and the size of its largest clique mino ...
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 is ...
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 P(G,t) has no zeros in the region bipartite, and thus computable in linear time using breadth-first search or depth-first search. More generally, the chromatic number and a corresponding coloring of
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 perfe ...
s can be computed in polynomial time using
semidefinite programming Semidefinite programming (SDP) is a subfield of convex optimization concerned with the optimization of a linear objective function (a user-specified function that the user wants to minimize or maximize) over the intersection of the cone of positive ...
. 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 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 Dynamic programming is both a mathematical optimization method and a computer programming method. The method was developed by Richard Bellman in the 1950s and has found applications in numerous fields, from aerospace engineering to economics. ...
and a bound on the number of
maximal independent set In graph theory, a maximal independent set (MIS) or maximal stable set is an independent set that is not a subset of any other independent set. In other words, there is no vertex outside the independent set that may join it because it is max ...
s, ''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^nn) 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 Contraction may refer to: Linguistics * Contraction (grammar), a shortened word * Poetic contraction, omission of letters for poetic reasons * Elision, omission of sounds ** Syncope (phonology), omission of sounds in a word * Synalepha, merged ...
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 William Thomas Tutte OC FRS FRSC (; 14 May 1917 – 2 May 2002) was an English and Canadian codebreaker and mathematician. During the Second World War, he made a brilliant and fundamental advance in cryptanalysis of the Lorenz cipher, a majo ...
's curiosity about which other graph properties satisfied this recurrence led him to discover a bivariate generalization of the chromatic polynomial, the
Tutte polynomial The Tutte polynomial, also called the dichromate or the Tutte–Whitney polynomial, is a graph polynomial. It is a polynomial in two variables which plays an important role in graph theory. It is defined for every undirected graph G and contai ...
. 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 numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...
, 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 an algorithm design paradigm for discrete and combinatorial optimization problems, as well as mathematical optimization. A branch-and-bound algorithm consists of a systematic enumeration of candidate solut ...
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) ar ...
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 In graph theory, a branch of mathematics, a crown graph on vertices is an undirected graph with two sets of vertices and and with an edge from to whenever . The crown graph can be viewed as a complete bipartite graph from which the edges ...
on ''n'' vertices can be 2-colored, but has an ordering that leads to a greedy coloring with n/2 colors. For
chordal graph In the mathematical area of graph theory, a chordal graph is one in which all cycles of four or more vertices have a ''chord'', which is an edge that is not part of the cycle but connects two vertices of the cycle. Equivalently, every induced c ...
s, and for special cases of chordal graphs such as
interval graph In graph theory, an interval graph is an undirected graph formed from a set of intervals on the real line, with a vertex for each interval and an edge between vertices whose intervals intersect. It is the intersection graph of the intervals. In ...
s and
indifference graph In graph theory, a branch of mathematics, an indifference graph is an undirected graph constructed by assigning a real number to each vertex and connecting two vertices by an edge when their numbers are within one unit of each other.. Indifference ...
s, 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 In the mathematical area of graph theory, a chordal graph is one in which all cycles of four or more vertices have a ''chord'', which is an edge that is not part of the cycle but connects two vertices of the cycle. Equivalently, every induced c ...
for the graph. The
perfectly orderable graph In graph theory, a perfectly orderable graph is a graph whose vertices can be ordered in such a way that a greedy coloring algorithm with that ordering optimally colors every induced subgraph of the given graph. Perfectly orderable graphs form a s ...
s generalize this property, but it is NP-hard to find a perfect ordering of these graphs. If the vertices are ordered according to their
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 mathematics ...
s, 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 In graph theory, the Grundy number or Grundy chromatic number of an undirected graph is the maximum number of colors that can be used by a greedy coloring strategy that considers the vertices of the graph in sequence and assigns each vertex its f ...
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 discre ...
one after another, expending a previously unused colour when needed. Once a new
vertex Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet * Vertex (computer graphics), a data structure that describes the positio ...
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 In graph theory, a maximal independent set (MIS) or maximal stable set is an independent set that is not a subset of any other independent set. In other words, there is no vertex outside the independent set that may join it because it is max ...
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 \mathcal(mn). DSatur and RLF are exact for bipartite, cycle, and
wheel graph A wheel is a circular component that is intended to rotate on an axle bearing. The wheel is one of the key components of the wheel and axle which is one of the six simple machines. Wheels, in conjunction with axles, allow heavy objects to be ...
s.


Parallel and distributed algorithms

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, scientific ...
s, graph coloring is closely related to the problem of
symmetry breaking In physics, symmetry breaking is a phenomenon in which (infinitesimally) small fluctuations acting on a system crossing a critical point decide the system's fate, by determining which branch of a bifurcation is taken. To an outside observe ...
. 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 et al. In a
symmetric graph In the mathematical field of graph theory, a graph is symmetric (or arc-transitive) if, given any two pairs of adjacent vertices and of , there is an automorphism :f : V(G) \rightarrow V(G) such that :f(u_1) = u_2 and f(v_1) = v_2. In oth ...
, a deterministic 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 i ...
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 In computer science, the iterated logarithm of n, written  n (usually read "log star"), is the number of times the logarithm function must be iteratively applied before the result is less than or equal to 1. The simplest formal definition i ...
, 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 graph In geometric graph theory, a unit disk graph is the intersection graph of a family of unit disks in the Euclidean plane. That is, it is a graph with one vertex for each disk in the family, and with an edge between two vertices whenever the corr ...
s by Schneider et al. 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 support ...
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, a problem is NP-complete when: # it is a problem for which the correctness of each solution can be verified quickly (namely, in polynomial time) and a brute-force search algorithm can find a solution by trying ...
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 graphs. On graphs with maximal degree 3 or less, however,
Brooks' theorem In graph theory, Brooks' theorem states a relationship between the maximum 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 vertices can be colored with ...
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, and it is possible to find such a coloring in polynomial time. The best known approximation algorithm 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. It is also NP-hard to color a 3-colorable graph with 4 colors and a ''k''-colorable graph with ''k''(log ''k'' ) / 25 colors for sufficiently large constant ''k''. 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 In computer science (particularly algorithmics), a polynomial-time approximation scheme (PTAS) is a type of approximation algorithm for optimization problems (most often, NP-hard optimization problems). A PTAS is an algorithm which takes an insta ...
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 In graph theory, an interval graph is an undirected graph formed from a set of intervals on the real line, with a vertex for each interval and an edge between vertices whose intervals intersect. It is the intersection graph of the intervals. In ...
, 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 In geometric graph theory, a unit disk graph is the intersection graph of a family of unit disks in the Euclidean plane. That is, it is a graph with one vertex for each disk in the family, and with an edge between two vertices whenever the corr ...
, so the coloring problem is 3-approximable.


Register allocation

A
compiler In computing, a compiler is a computer program that translates computer code written in one programming language (the ''source'' language) into another language (the ''target'' language). The name "compiler" is primarily used for programs tha ...
is a
computer program A computer program is a sequence or set of instructions in a programming language for a computer to execute. Computer programs are one component of software, which also includes documentation and other intangible components. A computer program ...
that translates one
computer language A computer language is a formal language used to communicate with a computer. Types of computer languages include: * Construction language – all forms of communication by which a human can specify an executable problem solution to a comput ...
into another. To improve the execution time of the resulting code, one of the techniques of compiler optimization is
register allocation In compiler optimization, register allocation is the process of assigning local automatic variables and expression results to a limited number of processor registers. Register allocation can happen over a basic block (''local register allocatio ...
, where the most frequently used values of the compiled program are kept in the fast processor registers. 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
pattern matching In computer science, pattern matching is the act of checking a given sequence of tokens for the presence of the constituents of some pattern. In contrast to pattern recognition, the match usually has to be exact: "either it will or will not be ...
, sports scheduling, designing seating plans, exam timetabling, the scheduling of taxis, and solving
Sudoku Sudoku (; ja, 数独, sūdoku, digit-single; originally called Number Place) is a logic-based, combinatorial number-placement puzzle. In classic Sudoku, the objective is to fill a 9 × 9 grid with digits so that each column, each row ...
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 mathematics that focuses on the appearance of order in a substructure given a structure of a known size. Problems in Ramsey theory typically ask ...
, 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 The theorem on friends and strangers is a mathematical theorem in an area of mathematics called Ramsey theory. Statement Suppose a party has six people. Consider any two of them. They might be meeting for the first time—in which case we will ...
, 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.


Other colorings

;
Adjacent-vertex-distinguishing-total coloring In graph theory, a total coloring is a coloring on the vertices and edges of a graph such that: (1). no adjacent vertices have the same color; (2). no adjacent edges have the same color; and (3). no edge and its endvertices are assigned the sam ...
: 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. ; Circular coloring : Motivated by task systems in which production proceeds in a cyclic way ;
Cocoloring In graph theory, a cocoloring of a graph ''G'' is an assignment of colors to the vertices such that each color class forms an independent set in ''G'' or in the complement of ''G''. The cochromatic number z(''G'') of ''G'' is the fewest colors ne ...
: An improper vertex coloring where every color class induces an independent set or a clique ;
Complete coloring In graph theory, a complete coloring is a vertex coloring in which every pair of colors appears on ''at least'' one pair of adjacent vertices. Equivalently, a complete coloring is minimal in the sense that it cannot be transformed into a proper ...
: 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 In graph theory, a distinguishing coloring or distinguishing labeling of a graph is an assignment of colors or labels to the vertices of the graph that destroys all of the nontrivial symmetries of the graph. The coloring does not need to be a ...
: An improper vertex coloring that destroys all the symmetries of the graph ;
Equitable coloring In graph theory, an area of mathematics, an equitable coloring is an assignment of colors to the vertices of an undirected graph, in such a way that *No two adjacent vertices have the same color, and *The numbers of vertices in any two color classe ...
: The sizes of color classes differ by at most one ;
Exact coloring In graph theory, an exact coloring is a (proper) vertex coloring in which every pair of colors appears on exactly one pair of adjacent vertices. That is, it is a partition of the vertices of the graph into disjoint independent sets such that, fo ...
: Every pair of colors appears on exactly one edge ;
Fractional coloring Fractional coloring is a topic in a young branch of graph theory known as fractional graph theory. It is a generalization of ordinary graph coloring. In a traditional graph coloring, each vertex in a graph is assigned some color, and adjacent ve ...
: 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 In graph theory, a harmonious coloring is a (proper) vertex coloring in which every pair of colors appears on ''at most'' one pair of adjacent vertices. It is the opposite of the complete coloring, which instead requires every color pairing to o ...
: Every pair of colors appears on at most one edge ;
Incidence coloring In graph theory, the act of coloring generally implies the assignment of labels to vertices, edges or faces in a graph. The incidence coloring is a special graph labeling where each incidence of an edge with a vertex is assigned a color under ce ...
: Each adjacent incidence of vertex and edge is colored with distinct colors ; 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 In mathematics, list edge-coloring is a type of graph coloring that combines list coloring and edge coloring. An instance of a list edge-coloring problem consists of a graph together with a list of allowed colors for each edge. A list edge-colori ...
: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 In graph theory, path coloring usually refers to one of two problems: * The problem of coloring a (multi)set of paths R in graph G, in such a way that any two paths of R which share an edge in G receive different colors. Set R and graph G are pro ...
: 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 In graph theory, a subcoloring is an assignment of colors to a graph's vertices such that each color class induces a vertex disjoint union of cliques. That is, each color class should form a cluster graph. The subchromatic number χS(''G'') of a ...
: An improper vertex coloring where every color class induces a union of cliques ;
Sum coloring In graph theory, a sum coloring of a graph is a labeling of its vertices by positive integers, with no two adjacent vertices having equal labels, that minimizes the sum of the labels. The minimum sum that can be achieved is called the chromatic sum ...
: The criterion of minimalization is the sum of colors ; Star coloring : Every 2-chromatic subgraph is a disjoint collection of
stars A star is an astronomical object comprising a luminous spheroid of plasma held together by its gravity. The nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night, but their immense distances from Earth ma ...
; 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 In graph theory, a T-Coloring of a graph G = (V, E), given the set ''T'' of nonnegative integers containing 0, is a function c: V(G) \to \N that maps each vertex to a positive integer (color) such that if ''u'' and ''w'' are adjacent then , c(u) - c ...
: Absolute value of the difference between two colors of adjacent vertices must not belong to fixed set ''T'' ;
Total coloring In graph theory, total coloring is a type of graph 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 edges, no adjacent vertices ...
:Vertices and edges are colored ; Centered coloring: Every connected
induced subgraph In the mathematical field of 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. Defini ...
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 graph A gain graph is a graph whose edges are labelled "invertibly", or "orientably", by elements of a group ''G''. This means that, if an edge ''e'' in one direction has label ''g'' (a group element), then in the other direction it has label ''g''  ...
s.


See also

*
Critical graph In graph theory, a critical graph is an undirected graph all of whose subgraphs have smaller chromatic number. In such a graph, every vertex or edge is a critical element, in the sense that its deletion would decrease the number of colors needed ...
*
Graph coloring game The graph coloring game is a mathematical game related to graph theory. Coloring game problems arose as game-theoretic versions of well-known graph coloring problems. In a coloring game, two players use a given set of colors to construct a coloring ...
*
Graph homomorphism In the mathematical field of graph theory, a graph homomorphism is a mapping between two graphs that respects their structure. More concretely, it is a function between the vertex sets of two graphs that maps adjacent vertices to adjacent verti ...
*
Hajós construction In graph theory, a branch of mathematics, the Hajós construction is an operation on graphs named after that may be used to construct any critical graph or any graph whose chromatic number is at least some given threshold. The construction Let ...
*
Mathematics of Sudoku The mathematics of Sudoku refers to the use of mathematics 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 S ...
*
Multipartite graph In graph theory, a part of mathematics, a -partite graph is a graph whose vertices are (or can be) partitioned into different independent sets. Equivalently, it is a graph that can be colored with colors, so that no two endpoints of an edge ...
*
Uniquely colorable graph In graph theory, a uniquely colorable graph is a -chromatic graph that has only one possible (proper) - coloring up to permutation of the colors. Equivalently, there is only one way to partition its vertices into independent sets and there i ...


Notes


References

* * * * * * (= ''Indag. Math.'' 13) * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *. *. * * * * * * * * * * * . Translated into English in ''Amer. Math. Soc. Translation'', 1952, .


External links


''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).

by Joseph Culberson (graph coloring programs)
''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 App
by Jose Antonio Martin H. {{DEFAULTSORT:Graph Coloring Coloring NP-complete problems NP-hard problems Computational problems in graph theory Extensions and generalizations of graphs