Combinatorics and Graph Theory with Mathematica. Evaluating the chromatic polynomial in variables at the points , 2, OEIS A , resulting in chromatic polynomial. The chromatic polynomial of an undirected graph , also denoted Biggs , p. The chromatic polynomial is multiplicative over graph components, so for a graph having connected components , , Interestingly, is equal to the number of acyclic orientations of Stanley

Contact the MathWorld Team. 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 , p. Furthermore, the coefficients alternate signs, and the coefficient of the st term is , where is the number of edges. The chromatic polynomial of a planar graph is related to the flow polynomial of its dual graph by. The following table summarizes the chromatic polynomials for some simple graphs. In fact, evaluating at integers still gives the numbers of -colorings.

## Chromatic Polynomial

Is cbrt 3 an irrational number? More precisely, if is the number of graph vertices of such a graphthen.

Evaluating at2, Here dhromatic the falling factorial. The chromatic polynomial of a disconnected graph is the product of the chromatic polynomials of its connected components. Except for special cases such as treesthe calculation of is exponential in the minimum number of edges in and the graph complement Skienap.

The chromatic polynomial of a graph in the variable can be determined in the Wolfram Language using ChromaticPolynomial [ g homewor, x ]. Hints help you try the next step on your own.

Wed May 15 Interestingly, is equal to the number of acyclic orientations of Stanley Practice online or make a printable study sheet.

# Chromatic Polynomial — from Wolfram MathWorld

Contact the MathWorld Team. Cambridge University Press, pp. Conjecture de Beraha pour les cycles French Jacqueline Zizi. For a graph with vertex count and connected components, homeqork chromatic polynomial is related to the rank polynomial and Tutte polynomial by. Collection of teaching and learning tools built by Wolfram education experts: The following table summarizes the recurrence relations for chromatic polynomials for some simple classes of graphs.

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. Combinatorics and Graph Theory with Mathematica. Walk through homework problems step-by-step from beginning to end. Furthermore, the coefficients alternate signs, and the coefficient of the st term iswhere polynomixl the number of edges.

For a graph on vertices that can be colored in ways with no colors, way with one color, OEIS A polynomjal, resulting in chromatic polynomial. The chromatic polynomial for a forest on vertices, edges, and with connected components is given by. Evaluating the chromatic polynomial in variables at the points2, The chromatic polynomial of an undirected graphalso denoted Biggsp.

The chromatic polynomial of a planar graph is related to the flow polynomial of its dual graph by. In fact, evaluating at integers still gives the numbers of -colorings. The following table summarizes the chromatic polynomials for some simple graphs. Chromatic Polynomials Jaime Rangel-Mondragon. Chromatic polynomials are not diagnostic for graph isomorphism, i.

The chromatic polynomial is multiplicative over graph components, so for a graph having connected components, The chromatic number of a graph gives the smallest number of colors polynomil which a graph can be colored, which is therefore the smallest positive integer such that Skienap.

Precomputed chromatic polynomials for many named graphs can be obtained using GraphData [ graph”ChromaticPolynomial” ][ z ]. The chromatic polynomial of a graph of order has degreewith leading coefficient 1 and constant term 0.

For example, the cubical graph has 1- 2-