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arXiv · 2506.00941

Interpreting the (signed) chromatic polynomial coefficients via hyperplane arrangements

Neha Goregaokar
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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.

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Neha Goregaokar. 2025-10-23. Interpreting the (signed) chromatic polynomial coefficients via hyperplane arrangements. https://arxiv.org/abs/2506.00941

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