Interpreting the (signed) chromatic polynomial coefficients via hyperplane arrangements

2025/06/01 by Goregaokar, Neha
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2506.00941

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.

Citations

• An identity for the coefficients of characteristic polynomials of hyperplane arrangements
• Combinatorial reciprocity for the chromatic polynomial and the chromatic\n symmetric function

Related

• Hyperoctahedral Eulerian Idempotents, Hodge Decompositions, and Signed\n Graph Coloring Complexes
• Acyclic Orientations and the Chromatic Polynomial of Signed Graphs
• Characteristic Polynomial of Certain Hyperplane Arrangements through Graph Theory
• Chromatic polynomials of signed graphs and dominating-vertex deletion formulae
• The chromatic number of a signed graph