The cyclic coloring complex of a complete k-uniform hypergraph

In this paper, we study the homology of the cyclic coloring complex of three different types of $k$-uniform hypergraphs. For the case of a complete $k$-uniform hypergraph, we show that the dimension of the $(n-k-1)^{st}$ homology group is given by a binomial coefficient. Further, we discuss a complex whose $r$-faces consist of all ordered set partitions $[B_1, ..., B_{r+2}]$ where none of the $B_i$ contain a hyperedge of the complete $k$-uniform hypergraph $H$ and where $1 \in B_1$. It is shown that the dimensions of the homology groups of this complex are given by binomial coefficients. As a consequence, this result gives the dimensions of the multilinear parts of the cyclic homology groups of \C[x_1,...,x_n]/ \{x_{i_1}...x_{i_k} \mid i_{1}...i_{k} is a hyperedge of $H \}$. For the other two types of hypergraphs, star hypergraphs and diagonal hypergraphs, we show that the dimensions of the homology groups of their cyclic coloring complexes are given by binomial coefficients as well

Hyperoctahedral Eulerian Idempotents, Hodge Decompositions, and Signed Graph Coloring Complexes

Phil Hanlon proved that the coefficients of the chromatic polynomial of a graph G are equal (up to sign) to the dimensions of the summands in a Hodge-type decomposition of the top homology of the coloring complex for G. We prove a type B analogue of this result for chromatic polynomials of signed graphs using hyperoctahedral Eulerian idempotents

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 Hn−3(Δ(G)) where the dimension of the jth component in the decomposition, H(j) n−3(Δ(G)), equals the absolute value of the coefficient of λj in the chromatic polynomial of G, χG(λ). Let H 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 λj inχH(λ) gives the Euler Characteristic of the jth 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 λj in χH(λ) equals the dimension of the jth Hodge piece of the Hodge decomposition of Δ(H). We also note that the Euler Characteristic of the jth Hodge subcomplex of the Hodge decomposition of the intersection of coloring complexes is given by the coefficient of jth term in the associated chromatic polynomial