The Hodge structure of the coloring complex of a hypergraph

Let G be a simple graph with n vertices. The coloring complex Δ ( G ) was defined by Steingrímsson, 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 ) . be a hypergraph with n vertices. In this paper, we define the coloring complex of a hypergraph, Δ ( H ) , and show that the coefficient of λ j in χ H ( λ ) gives the Euler Characteristic of the j th Hodge subcomplex of the Hodge decomposition of Δ ( H ) . We also examine conditions on a hypergraph, H , for which its Hodge subcomplexes are Cohen–Macaulay, and thus where the absolute value of the coefficient of λ j in χ H ( λ ) equals the dimension of the j th Hodge piece of the Hodge decomposition of Δ ( H ) . We also note that the Euler Characteristic of the j th Hodge subcomplex of the Hodge decomposition of the intersection of coloring complexes is given by the coefficient of j th term in the associated chromatic polynomial.