Séminaire Lotharingien de Combinatoire, 93B.32 (2025), 12 pp.
Neha Goregaokar
Interpreting the 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. 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. We use this generalization to give a new proof of the formula for the chromatic polynomial of a natural unit interval graph.
Received: November 15, 2024.
Accepted: February 15, 2025.
Final version: April 1, 2025.
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