Chromatic Polynomial

The chromatic polynomial of an undirected graph , also denoted (Biggs 1973, p. 106) and (Godsil and Royle 2001, p. 358), is a polynomial which encodes the number of distinct ways to color 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 variables 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.

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, x]. 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 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 the recurrence relations for chromatic polynomials for some simple classes of graphs.

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, and the coefficient of the st term is , where is the number of edges. Interestingly, is equal to the number of acyclic orientations of (Stanley 1973).

Except for special cases (such as trees), the calculation of is exponential in the minimum number of edges in and the graph complement (Skiena 1990, p. 211), and calculating the chromatic polynomial of a graph is at least an NP-complete problem (Skiena 1990, pp. 211-212).

Tutte (1970) showed that the chromatic polynomial of a planar triangulation of a sphere possess a root close to (OEIS A104457), where is the golden ratio. More precisely, if is the number of graph vertices of such a graph , then

(7)

(Tutte 1970, Le Lionnais 1983).

Read (1968) conjectured that, for any chromatic polynomial

(8)

there does not exist a such that and (Skiena 1990, p. 221).