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

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.