Approximating the Chromatic Polynomial

TL;DR

This paper introduces two algorithms that approximate the coefficients of the chromatic polynomial of a graph, enabling analysis of larger or more complex graphs beyond previous computational limits.

Contribution

The authors develop and implement two novel algorithms based on Knuth's search tree estimation method for approximating chromatic polynomial coefficients.

Findings

Algorithms provide accurate approximations compared to true polynomials.

Error rates are lower than previous approximation methods.

Applicable to larger and more complex graphs than prior techniques.

Abstract

Chromatic polynomials are important objects in graph theory and statistical physics, but as a result of computational difficulties, their study is limited to graphs that are small, highly structured, or very sparse. We have devised and implemented two algorithms that approximate the coefficients of the chromatic polynomial $P(G,x)$, where $P(G,k)$ is the number of proper $k$-colorings of a graph $G$ for $k\in\mathbb{N}$. Our algorithm is based on a method of Knuth that estimates the order of a search tree. We compare our results to the true chromatic polynomial in several known cases, and compare our error with previous approximation algorithms.