Interpreting the (signed) chromatic polynomial coefficients via hyperplane 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.
Keywords
Cite
@article{arxiv.2506.00941,
title = {Interpreting the (signed) chromatic polynomial coefficients via hyperplane arrangements},
author = {Neha Goregaokar},
journal= {arXiv preprint arXiv:2506.00941},
year = {2025}
}
Comments
30 pages, 12 figures. v2: Added Section 6 generalizing to subarrangements of Type B Coxeter arrangement and signed graphs. Extended abstract of this paper was accepted for FPSAC 2025