The
chromatic polynomial is a
polynomialIn mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
studied in
algebraic graph theoryAlgebraic 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...
, a branch of
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
. It counts the number of
graph coloringIn graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices share the...
s as a function of the number of colors and was originally defined by
George David Birkhoff-External links:* − from National Academies Press, by Oswald Veblen....
to attack the four color problem. It was generalised to the
Tutte polynomialThe Tutte polynomial, also called the dichromate or the Tutte–Whitney polynomial, is a polynomial in two variables which plays an important role in graph theory, a branch of mathematics and theoretical computer science...
by H. Whitney and
W. T. TutteWilliam Thomas Tutte, OC, FRS, known as Bill Tutte, was a British, later Canadian, codebreaker and mathematician. During World War II he made a brilliant and fundamental advance in Cryptanalysis of the Lorenz cipher, a major German code system, which had a significant impact on the Allied...
, linking it to the
Potts modelIn statistical mechanics, the Potts model, a generalization of the Ising model, is a model of interacting spins on a crystalline lattice. By studying the Potts model, one may gain insight into the behaviour of ferromagnets and certain other phenomena of solid state physics...
of
statistical physicsStatistical physics is the branch of physics that uses methods of probability theory and statistics, and particularly the mathematical tools for dealing with large populations and approximations, in solving physical problems. It can describe a wide variety of fields with an inherently stochastic...
.
History
George David Birkhoff-External links:* − from National Academies Press, by Oswald Veblen....
introduced the chromatic polynomial in 1912, defining it only for
planar graphIn 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...
s, in an attempt to prove the
four color theoremIn mathematics, the four color theorem, or the four color map theorem states that, given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colors are required to color the regions of the map so that no two adjacent regions have the same color...
. If
denotes the number of proper colorings of
with
colors then one could establish the four color theorem by showing
for all planar graphs
. In this way he hoped to apply the powerful tools of analysis and algebra for studying the roots of polynomials to the combinatorial coloring problem.
Hassler WhitneyHassler Whitney was an American mathematician. He was one of the founders of singularity theory, and did foundational work in manifolds, embeddings, immersions, and characteristic classes.-Work:...
generalised Birkhoff’s polynomial from the planar case to general graphs in 1932. In 1968 Read asked which polynomials are the chromatic polynomials of some graph, a question that remains open, and introduced the concept of chromatically equivalent graphs. Today, chromatic polynomials are one of the central objects of
algebraic graph theoryAlgebraic 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...
Definition
The chromatic polynomial of a graph counts the number of its proper vertex colorings. It is commonly denoted
,
, or
, and sometimes in the form
, where it is understood that for fixed
the function is a polynomial in
, the number of colors.
For example, the
path graphIn the mathematical field of graph theory, a path graph or linear graph is a particularly simple example of a tree, namely a tree with two or more vertices that is not branched at all, that is, contains only vertices of degree 2 and 1...
on 3 vertices cannot be colored at all with 0 or 1 colors. With 2 colors, it can be colored in 2 ways. With 3 colors, it can be colored in 12 ways.
Available colors  |
0 |
1 |
2 |
3 |
Number of colorings  |
0 |
0 |
2 |
12 |
The chromatic polynomial is defined as the unique interpolating polynomial of degree
through the points
for
, where
is the number of vertices in
.
For the example graph,
, and indeed
.
The chromatic polynomial includes at least as much information about the colorability of
as does the chromatic number. Indeed, the chromatic number is the smallest positive integer that is not a root of the chromatic polynomial,
-
Examples
Chromatic polynomials for certain graphs
Triangle  |
 |
Path graphIn the mathematical field of graph theory, a path graph or linear graph is a particularly simple example of a tree, namely a tree with two or more vertices that is not branched at all, that is, contains only vertices of degree 2 and 1...
 |
 |
Complete graphIn 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.-Properties:...
 |
 |
Tree with  vertices |
 |
CycleIn 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 connected in a closed chain. The cycle graph with n vertices is called Cn...
 |
 |
Petersen graphIn the mathematical field of graph theory, the Petersen graph is an undirected graph with 10 vertices and 15 edges. It is a small graph that serves as a useful example and counterexample for many problems in graph theory. The Petersen graph is named for Julius Petersen, who in 1898 constructed it...
|
 |
Properties
For fixed
on
vertices, the chromatic polynomial
is in fact a polynomial; it has degree
. Nonisomorphic graphs may have the same chromatic polynomial.
By definition, evaluating the chromatic polynomial in
yields the number of
-colorings of
for
…
. Perhaps surprisingly, the same holds for any
, and besides,
yields the number of
acyclic orientationsIn mathematics and computer science, a directed acyclic graph , is a directed graph with no directed cycles. That is, it is formed by a collection of vertices and directed edges, each edge connecting one vertex to another, such that there is no way to start at some vertex v and follow a sequence of...
of
. Furthermore,
the derivative evaluated at 1,
equals the
chromatic invariant 
up to sign.
If
has
vertices,
edges, and
componentsIn graph theory, a connected component of an undirected graph is a subgraph in which any two vertices are connected to each other by paths, and which is connected to no additional vertices. For example, the graph shown in the illustration on the right has three connected components...
…,
, then
- The coefficient of
in 
is 1.
- The coefficient of
in 
is .
- The coefficients of
are all nonzero.
- The coefficient of
is nonzero.
-
- The coefficients of every chromatic polynomial alternate in signs.
A graph G with
vertices is a tree if and only if
.
Chromatic equivalence
Two graphs are said to be chromatically equivalent if they have the same chromatic polynomial. Isomorphic graphs have the same chromatic polynomial, but nonisomorphic graphs can be chromatically equivalent. For example, all trees on
vertices have the same chromatic polynomial; in particular,
is the chromatic polynomial of both the
claw graphIn graph theory, an area of mathematics, a claw-free graph is a graph that does not have a claw as an induced subgraph.A claw is another name for the complete bipartite graph K1,3...
and the
path graphIn the mathematical field of graph theory, a path graph or linear graph is a particularly simple example of a tree, namely a tree with two or more vertices that is not branched at all, that is, contains only vertices of degree 2 and 1...
on 4 vertices.
Chromatic uniqueness
A graph is chromatically unique if it is determined by its chromatic polynomial, up to isomorphism. In other words,
is chromatically unique if
implies that
and
are isomorphic. All
Cycle graphIn 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 connected in a closed chain. The cycle graph with n vertices is called Cn...
s are chromatically unique.
Chromatic roots
A root (or
zero) of a chromatic polynomial, called a “chromatic root”, is a value
where
. Chromatic roots have been very well studied, in fact, Birkhoff’s original motivation for defining the chromatic polynomial was to show that for planar graphs,
for
≥ 4. This would have established the
four color theoremIn mathematics, the four color theorem, or the four color map theorem states that, given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colors are required to color the regions of the map so that no two adjacent regions have the same color...
.
No graph can be 0-colored, so 0 is always a chromatic root. Only edgeless graphs can be 1-colored, so 1 is a chromatic root for every graph with at least an edge. On the other hand, except for these two points, no graph can have a chromatic root at a real number smaller than or equal to 32/27. A result of Tutte connects the
golden ratioIn mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.61803398874989...
with the study of chromatic roots, showing that chromatic roots exist very close to
:
If
is a planar triangulation of a sphere then
≤
. While the real line thus has large parts that contain no chromatic roots for any graph, every point in the
complex planeIn mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...
is arbitrarily close to a chromatic root in the sense that there exists an infinite family of graphs whose chromatic roots are dense in the complex plane.
Algorithms
cellspacing="5" style="width: 22em; text-align: left; font-size: 88%; line-height: 1.5em; ">
| Chromatic polynomial |
| Input | Graph  with  vertices. |
| Output | Coefficients of  |
| Running time |  for some constant  |
| Complexity | #P In computational complexity theory, the complexity class #P is the set of the counting problems associated with the decision problems in the set NP. More formally, #P is the class of function problems of the form "compute ƒ," where ƒ is the number of accepting paths of a nondeterministic...
-hard |
| Reduction from | #3SAT |
cellspacing="5" style="width: 22em; text-align: left; font-size: 88%; line-height: 1.5em; ">
| #k-colorings |
| Input | Graph  with  vertices. |
| Output |  |
| Running time | In P for  .  for  . Otherwise
 for some constant  |
| Complexity | #P In computational complexity theory, the complexity class #P is the set of the counting problems associated with the decision problems in the set NP. More formally, #P is the class of function problems of the form "compute ƒ," where ƒ is the number of accepting paths of a nondeterministic...
-hard unless  |
| Approximability | No FPRAS for 
Computational problems associated with the chromatic polynomial include
The first problem is more general because if we knew the coefficients of  we could evaluate it at any point in polynomial time because the degree is  . The difficulty of the second type of problem depends strongly on the value of  and has been intensively studied in computational complexityComputational Complexity may refer to:*Computational complexity theory*Computational Complexity...
.
Efficient algorithms
For some basic graph classes, closed formulas for the chromatic polynomial are known. For instance this is true for trees and cliques, as listed in the table above.
Polynomial time algorithms are known for computing the chromatic polynomial for wider classes of graphs, including chordal graphIn the mathematical area of graph theory, a graph is chordal if each of its cycles of four or more nodes has a chord, which is an edge joining two nodes that are not adjacent in the cycle. An equivalent definition is that any chordless cycles have at most three nodes...
s and graphs of bounded clique-widthIn graph theory, the clique-width of a graph G is the minimum number of labels needed to construct G by means of the following 4 operations :#Creation of a new vertex v with label i...
. The latter class includes cographIn graph theory, a cograph, or complement-reducible graph, or P4-free graph, is a graph that can be generated from the single-vertex graph K1 by complementation and disjoint union...
s and graphs of bounded tree-width, such as outerplanar graphIn graph theory, an undirected graph is an outerplanar graph if it can be drawn in the plane without crossings in such a way that all of the vertices belong to the unbounded face of the drawing. That is, no vertex is totally surrounded by edges...
s.
Deletion–contraction
A recursive way of computing the chromatic polynomial is based on edge contractionIn graph theory, an edge contraction is an operation which removes an edge from a graph while simultaneously merging together the two vertices it previously connected. Edge contraction is a fundamental operation in the theory of graph minors...
: for a pair of vertices  and  the graph  is obtained by merging the two vertices and removing any edges between them.
Then the chromatic polynomial satisfies the recurrence relation
where  and  are adjacent vertices and  is the graph with the edge  removed. Equivalently,
if  and  are not adjacent and  is the graph with the edge  added. In the first form, the recurrence terminates in a collection of empty graphs. In the second form, it terminates in a collection of complete graphs. These recurrences are also called the Fundamental Reduction Theorem
Tutte’s curiosity about which other graph properties satisfied such recurrences led him to discover a bivariate generalization of the chromatic polynomial, the Tutte polynomialThe Tutte polynomial, also called the dichromate or the Tutte–Whitney polynomial, is a polynomial in two variables which plays an important role in graph theory, a branch of mathematics and theoretical computer science...
.
The expressions give rise to a recursive procedure, called the deletion–contraction algorithm, which forms the basis of many algorithms for graph coloring. The ChromaticPolynomial function in the computer algebra system MathematicaMathematica is a computational software program used in scientific, engineering, and mathematical fields and other areas of technical computing...
uses the second recurrence if the graph is dense, and the first recurrence if the graph is sparse The worst case running time of either formula satisfies the same recurrence relation as the Fibonacci numbers, so in the worst case, the algorithm runs in time within a polynomial factor of  . The analysis can be improved to within a polynomial factor of the number  of spanning treesIn the mathematical field of graph theory, a spanning tree T of a connected, undirected graph G is a tree composed of all the vertices and some of the edges of G. Informally, a spanning tree of G is a selection of edges of G that form a tree spanning every vertex...
of the input graph.
In practice, branch and boundBranch and bound is a general algorithm for finding optimal solutions of various optimization problems, especially in discrete and combinatorial optimization...
strategies and graph isomorphismIn abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
rejection are employed to avoid some recursive calls, the running time depends on the heuristic used to pick the vertex pair.
Computational complexity
The problem of computing the number of 3-colorings of a given graph is a canonical example of a #PIn computational complexity theory, the complexity class #P is the set of the counting problems associated with the decision problems in the set NP. More formally, #P is the class of function problems of the form "compute ƒ," where ƒ is the number of accepting paths of a nondeterministic...
-complete problem, so the problem of computing the coefficients of the chromatic polynomial is #P-hard. Similarly, evaluating  for given  is #P-complete.
On the other hand, for  it is easy to compute  , so the corresponding problems are polynomial-time computable. For integers  the problem is #P-hard, which is established similar to the case  . In fact, it is known that  is #P-hard for all  (including negative integers and non-integers) except for the three “easy points” Thus, from the perspective of #P-hardness, the complexity of computing the chromatic polynomial is completely understood.
In the expansion  , the coefficient  is always equal to 1, and several other properties of the coefficients are known. This raises the question if some of the coefficients are easy to compute. However the computational problem of computing the  th coefficient of  for fixed  for given graph  is hard for #P.
No approximation algorithms for computing  are known for any  except for the three easy points. At the integer points  , the corresponding decision problemIn computability theory and computational complexity theory, a decision problem is a question in some formal system with a yes-or-no answer, depending on the values of some input parameters. For example, the problem "given two numbers x and y, does x evenly divide y?" is a decision problem...
of deciding if a given graph can be  -colored is NP-hardNP-hard , in computational complexity theory, is a class of problems that are, informally, "at least as hard as the hardest problems in NP". A problem H is NP-hard if and only if there is an NP-complete problem L that is polynomial time Turing-reducible to H...
. Such problems cannot be approximated to any multiplicative factor by a bounded-error probabilistic algorithm unless NP = RP, because any multiplicative approximation would distinguish the values 0 and 1, effectively solving the decision version in bounded-error probabilistic polynomial time. In particular, under the same assumption, this rules out the possibility of a fully polynomial time randomised approximation scheme (FPRAS). For other points, more complicated arguments are needed, and the question is the focus of active research. , it is known that there is no FPRAS for computing  for any  unless NP = RP holds.
External links
- PlanetMath
PlanetMath is a free, collaborative, online mathematics encyclopedia. The emphasis is on rigour, openness, pedagogy, real-time content, interlinked content, and also community of about 24,000 people with various maths interests. Intended to be comprehensive, the project is hosted by the Digital...
Chromatic polynomial
- Code for computing Tutte, Chromatic and Flow Polynomials by Gary Haggard, David J. Pearce and Gordon Royle: http://www.ecs.vuw.ac.nz/~djp/tutte/
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