#### Title

The Hodge structure of the coloring complex of a hypergraph

#### Document Type

Article

#### Publication Date

10-2011

#### Publication Title

Discrete Mathematics

#### Abstract

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.

#### Volume

311

#### Issue

28

#### First Page

2164

#### Last Page

2173

#### DOI

https://doi.org/10.1016/j.disc.2011.06.034

#### ISSN

0012-365X

#### Repository Citation

Long, J.; Rundell, S.; The Hodge Structure of the Coloring Complex of a Hypergraph, Discrete Mathematics, Vol. 311, Iss. 20, 28 October 2011, 2164–2173. Available at dx.doi.org/10.1016/j.disc.2011.06.034