The Amazing Chromatic Polynomial

TL;DR

The paper surveys the chromatic polynomial of a graph, exploring its properties, applications in various mathematical contexts, and connections to other areas like algebraic geometry and symmetric functions.

Contribution

It provides a comprehensive overview of the chromatic polynomial's properties, applications, and its surprising appearances in different mathematical fields.

Findings

Chromatic polynomial counts proper colorings of graphs.

Connections to acyclic orientations, hyperplane arrangements, and increasing forests.

Links to symmetric functions and algebraic geometry.

Abstract

Let G be a combinatorial graph with vertices V and edges E. A proper coloring of G is an assignment of colors to the vertices such that no edge connects two vertices of the same color. These are the colorings considered in the famous Four Color Theorem. It turns out that the number of proper colorings of G using t colors is a polynomial in t, called the chromatic polynomial of G. This polynomial has many wonderful properties. It also has the surprising habit of appearing in contexts which, a priori, have nothing to do with graph coloring. We will survey three such instances involving acyclic orientations, hyperplane arrangements, and increasing forests. In addition, connections to symmetric functions and algebraic geometry will be mentioned.