The chromatic polynomial is a 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 attack the four color problem. It was generalised to the Tutte polynomial by H. Whitney and W. T. Tutte, linking it to the Potts model of statistical physics.
George David Birkhoff introduced the chromatic polynomial in 1912, defining it only for planar graphs, in an attempt to prove the four color theorem. If denotes the number of proper colorings of G with k colors then one could establish the four color theorem by showing for all planar graphs G. 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 Whitney 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 theory.
The chromatic polynomial of a graph G counts the number of its proper vertex colorings. It is commonly denoted , , , and which we will use from now on.
For example, the path graph 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.
|Number of colorings||0||0||2||12|
For a graph G with n vertices, the chromatic polynomial is defined as the unique interpolating polynomial of degree at most n through the points
If G does not contain any vertex with a self-loop, then the chromatic polynomial is a monic polynomial of degree exactly n. In fact for the above example we have:
The chromatic polynomial includes at least as much information about the colorability of G as does the chromatic number. Indeed, the chromatic number is the smallest positive integer that is not a root of the chromatic polynomial,
|Any tree on n vertices|
For fixed G on n vertices, the chromatic polynomial is in fact a polynomial of degree n. By definition, evaluating the chromatic polynomial in yields the number of k-colorings of G for . The same hold for k > n.
The derivative evaluated at 1, equals the chromatic invariant, , up to sign.
If G has n vertices, m edges, and c components , then
- The coefficients of are zeros.
- The coefficients of are all non-zero.
- The coefficient of in is 1.
- The coefficient of in is .
- The coefficients of every chromatic polynomial alternate in signs.
- The absolute values of coefficients of every chromatic polynomial form a log-concave sequence.
A graph G with n vertices is a tree if and only if
Two graphs are said to be chromatically equivalent if they have the same chromatic polynomial. Isomorphic graphs have the same chromatic polynomial, but non-isomorphic graphs can be chromatically equivalent. For example, all trees on n vertices have the same chromatic polynomial:
A graph is chromatically unique if it is determined by its chromatic polynomial, up to isomorphism. In other words, G is chromatically unique, then would imply that G and H are isomorphic.
A root (or zero) of a chromatic polynomial, called a “chromatic root”, is a value x 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 x ≥ 4. This would have established the four color theorem.
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 ratio 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 plane 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.
|Input||Graph G with n vertices.|
|Running time||for some constant|
|Input||Graph G with n vertices.|
In P for . for . Otherwisefor some constant
|Approximability||No FPRAS for|
Computational problems associated with the chromatic polynomial include
- finding the chromatic polynomial of a given graph G;
- evaluating at a fixed point k for given G.
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 n. The difficulty of the second type of problem depends strongly on the value of k and has been intensively studied in computational complexity. When k is a natural number, this problem is normally viewed as computing the number of k-colorings of a given graph. For example, this includes the problem #3-coloring of counting the number of 3-colorings, a canonical problem in the study of complexity of counting, complete for the counting class #P.
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 graphs and graphs of bounded clique-width. The latter class includes cographs and graphs of bounded tree-width, such as outerplanar graphs.
A recursive way of computing the chromatic polynomial is based on edge contraction: 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 polynomial.
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 Mathematica 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
on a graph with n vertices and m edges. The analysis can be improved to within a polynomial factor of the number of spanning trees of the input graph. In practice, branch and bound strategies and graph isomorphism rejection are employed to avoid some recursive calls, the running time depends on the heuristic used to pick the vertex pair.
There is a natural geometric perspective on graph colorings by observing that, as an assignment of natural numbers to each vertex, a graph coloring is a vector in the integer lattice. Since two vertices and being given the same color is equivalent to the ’th and ’th coordinate in the coloring vector being equal, each edge can be associated with a hyperplane of the form . The collection of such hyperplanes for a given graph is called its graphic arrangement. The proper colorings of a graph are those lattice points which avoid forbidden hyperplanes. Restricting to a set of colors, the lattice points are contained in the cube . In this context the chromatic polynomial counts the number of lattice points in the -cube that avoid the graphic arrangement.
The problem of computing the number of 3-colorings of a given graph is a canonical example of a #P-complete problem, so the problem of computing the coefficients of the chromatic polynomial is #P-hard. Similarly, evaluating for given G 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 x (including negative integers and even all complex numbers) 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 ar for a fixed r and a given graph G is #P-hard.
No approximation algorithms for computing are known for any x except for the three easy points. At the integer points , the corresponding decision problem of deciding if a given graph can be k-colored is NP-hard. 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. As of 2008[update], it is known that there is no FPRAS for computing for any x > 2, unless NP = RP holds.
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