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

Let $G$ be a simple graph with $n$ vertices. The coloring complex $Δ(G)$ was defined by Steingr\'ımsson, and the homology of $Δ(G)$ 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 $H_{n-3}(Δ(G))$ where the dimension of the $j^{th}$ component in the decomposition, $H_{n-3}^{(j)}(Δ(G))$, equals the absolute value of the coefficient of $λ^{j}$ in the chromatic polynomial of $G$, $χ_{G}(λ)$. 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.