Chromatic Polynomial

The chromatic polynomial of a finite simple graph with at least one graph vertex, also denoted (Biggs 1993, p. 106) and (Godsil and Royle 2001, p. 358), is a polynomial which counts proper colorings of the vertices of (where colorings are counted as distinct even if they differ only by permutation of colors). For a graph on vertices that can be colored in ways with no colors, way with one color, ..., and ways with colors, the chromatic polynomial of is defined as the unique Lagrange interpolating polynomial of degree through the points , , ..., . Evaluating the chromatic polynomial in the variable at the points , 2, ..., then recovers the numbers of 1-, 2-, ..., and -colorings. In fact, evaluating at integers still gives the numbers of -colorings.

The chromatic polynomial is called the "chromial" for short by Bari (1974).

The chromatic number of a graph gives the smallest number of colors with which a graph can be colored, which is therefore the smallest positive integer such that (Skiena 1990, p. 211).

For example, the cubical graph has 1-, 2-, ... k-coloring counts of 0, 2, 114, 2652, 29660, 198030, 932862, 3440024, ... (OEIS A140986), resulting in chromatic polynomial

(1)

Evaluating at , 2, ... then gives 0, 2, 114, 2652, 29660, 198030, 932862, 3440024, ... as expected.

The number of minimum vertex colorings of a graph is given by , where is the chromatic polynomial of and is its chromatic number.

A root of a chromatic polynomial is known as a chromatic root and an interval containing no chromatic root is called a chromatic root-free interval.

The chromatic polynomial of a graph in the variable can be determined in the Wolfram Language using ChromaticPolynomial[g, z]. Precomputed chromatic polynomials for many named graphs can be obtained using GraphData[graph, "ChromaticPolynomial"][z].

The chromatic polynomial is multiplicative over graph components, so for a graph having connected components , , ..., the chromatic polynomial of itself is given by

(2)

The chromatic polynomial for a forest on vertices, edges, and with connected components is given by

(3)

For a graph with vertex count and connected components, the chromatic polynomial is related to the rank polynomial and Tutte polynomial by

			(4)
			(5)

(extending Biggs 1993, p. 106). The chromatic polynomial of a connected planar graph is related to the flow polynomial of its dual graph by

(6)

Chromatic polynomials are not diagnostic for graph isomorphism, i.e., two nonisomorphic graphs may share the same chromatic polynomial. A graph that is determined by its chromatic polynomial is said to be a chromatically unique graph; nonisomorphic graphs sharing the same chromatic polynomial are said to be chromatically equivalent.

Chromatic polynomials of the ladder graph and grid graph are considered by Yadav et al. (2024). The following table summarizes the chromatic polynomials for some simple graphs. Here is the falling factorial.

graph	chromatic polynomial
barbell graph
book graph
centipede graph
complete graph
cycle graph
gear graph
helm graph
ladder graph
ladder rung graph
Möbius ladder
pan graph
path graph
prism graph
star graph
sun graph
sunlet graph
triangular honeycomb rook graph
web graph
wheel graph

The following table summarizes recurrence relations for chromatic polynomials of some families of graphs. Here is the chromatic polynomial of the member with parameter , and each recurrence applies when all its subscripts lie in the corresponding family's parameter range.

graph	order	recurrence
antiprism graph	4
barbell graph	1
book graph	1
centipede graph	1
complete graph	1
cycle graph	2
gear graph	2
helm graph	2
ladder graph	1
ladder rung graph	1
Möbius ladder	4
pan graph	2
path graph	1
prism graph	4
star graph	1
sunlet graph	2
web graph	4
wheel graph	2

The chromatic polynomial of a disconnected graph is the product of the chromatic polynomials of its connected components. The chromatic polynomial of a graph of order has degree , with leading coefficient 1 and constant term 0. Furthermore, the coefficients alternate signs, with zero coefficients allowed. The coefficient of is , where is the number of edges. Interestingly, is equal to the number of acyclic orientations of (Stanley 1973).

Deletion-contraction algorithms for calculating have exponential worst-case running time (Skiena 1990, pp. 211-212). Computing the chromatic polynomial is an NP-hard problem, since its value at decides whether is 3-colorable, an NP-complete problem.

Tutte (1970) proved an upper bound on the absolute value of the chromatic polynomial of a planar graph that triangulates a sphere. If such a graph has vertices, then

(7)

where is the golden ratio and (OEIS A104457; Tutte 1970; Le Lionnais 1983, p. 46).

Read (1968) conjectured that, for any chromatic polynomial

(8)

there does not exist a such that and (Skiena 1990, p. 221). Huh (2012) proved that, for every finite simple graph, the absolute values of the coefficients of its chromatic polynomial form a logarithmically concave sequence with no internal zeros and hence a unimodal sequence, proving Read's conjecture.