A Note on the Hodge Structure of the Intersection of Coloring Complexes

TL;DR

This paper explores the Hodge structure of intersections of coloring complexes in graphs, linking chromatic polynomial coefficients to the Euler characteristic of Hodge subcomplexes, extending prior homology results.

Contribution

It establishes a connection between chromatic polynomial coefficients and the Euler characteristics of Hodge subcomplexes in intersections of coloring complexes.

Findings

Chromatic polynomial coefficients determine Hodge subcomplex Euler characteristics.

The work extends homology results to Hodge structures in coloring complex intersections.

Provides a new topological interpretation of chromatic polynomial coefficients.

Abstract

Let $G$ be a simple graph with $n$ vertices. The coloring complex $\Delta(G)$ was defined by Steingr\'{\i}msson, and the homology of $\Delta(G)$ was shown to be nonzero only in dimension $n-3$ by Jonsson. Hanlon recently showed that the Eulerian idempotents provide a decomposition of the homology group $H_{n-3}(\Delta(G))$ where the dimension of the $j^{th}$ component in the decomposition, $H_{n-3}^{(j)}(\Delta(G))$, equals the absolute value of the coefficient of $\lambda^{j}$ in the chromatic polynomial of $G$, $\chi_{G}(\lambda)$. Jonsson recently studied the topology of intersections of coloring complexes. In this note, we show that the coefficient of the ${j}^{th}$ term in the chromatic polynomial of the intersection of coloring complexes gives the Euler Characteristic of the $j^{th}$ Hodge subcomplex of the Hodge decomposition of the intersection of coloring complexes.