Interpreting the (signed) chromatic polynomial coefficients via hyperplane arrangements

TL;DR

This paper provides a new interpretation of chromatic polynomial coefficients using hyperplane arrangements, extending previous results to signed graphs and specific arrangements like the braid arrangement.

Contribution

It offers an alternative proof for the chromatic polynomial coefficients and generalizes the interpretation to signed graphs and certain graphical arrangements.

Findings

New combinatorial interpretation for chromatic polynomial coefficients

Extension of interpretation to signed graphs

Application to braid and unit interval graph arrangements

Abstract

A recent result of Lofano and Paolini expresses the characteristic polynomial of a real hyperplane arrangement in terms of a projection statistic on the regions of the arrangement. We use this result to give an alternative proof for Greene and Zaslavsky's interpretation for the coefficients of the chromatic polynomial of a graph and further generalize this interpretation to signed graphs. We also show that this projection statistic has a nice combinatorial interpretation in the case of the braid arrangement, which generalizes to graphical arrangements of natural unit interval graphs.