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
