Interpreting the (signed) chromatic polynomial coefficients via hyperplane arrangements

TL;DR

This work provides a unified framework to interpret the coefficients of chromatic and characteristic polynomials via a projection statistic on regions of real hyperplane arrangements. Building on the Lofano–Paolini/Kabluchko projection formula, it specializes to graphical and Type $B$ subarrangements, yielding concrete combinatorial meanings: for braid arrangements the statistic equals RLmin of the labeling permutation; for graphical arrangements it counts source components of the labeling acyclic orientation; for natural unit interval graphs it furnishes a product form of the chromatic polynomial; and for signed/B-type arrangements it extends to signed source components and symmetric acyclic orientations. These results give alternate proofs of Greene–Zaslavsky’s interpretations and unify them under a single geometric-combinatorial mechanism, with explicit extensions to symmetric graphs and $B$-graphical arrangements. The approach not only clarifies existing interpretations but also provides new, tractable forms for chromatic polynomials in special graph families. Overall, the paper deepens the link between hyperplane geometry and graph colorings, offering practical computational formulas and a broadened combinatorial perspective.

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.

Figures (12)

• Figure 1: A graph $G$ and the corresponding graphical arrangement $\mathcal{A}_G$.
• Figure 2: The source components of the 6 acyclic orientations of $K_3$.
• Figure 3: A hyperplane arrangement in $\mathbb{R}^2$ with a point $v \in \mathbb{R}^2$ and the projection from $v$ to some regions. Here, $\textup{pd}_v(R_1) = 1$, $\textup{pd}_v(R_2) = 0$, and $\textup{pd}_v(R_3) = 1$.
• Figure 4: $\mathcal{A}_3$ with the faces labeled by ordered partitions of $[3]$.
• Figure 5: $\mathcal{B}_3$ with projection from a point $v$ as in Equation \ref{['vector']} onto the regions, with the regions labeled by permutations (in blue) with right-to-left minima marked by dots, and dimension of projection/number of right-to-left minima (in green).

Theorems & Definitions (71)

• Definition 2.1
• Definition 2.2
• Theorem 2.3
• Definition 2.4
• Theorem 2.5
• Definition 3.1
• Definition 3.2
• Theorem 3.3
• Remark 3.4
• Lemma 3.5