Let G be a simple graph with n vertices. The coloring complex ∆(G) was defined by Steingr ́ımsson, and
the homology of ∆(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(∆(G)) where the dimension
of the jth component in the decomposition, H(j)
n−3(∆(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, ∆(H), and show that the coefficient of λj in χH (λ) gives the Euler Characteristic of the
jth Hodge subcomplex of the Hodge decomposition of ∆(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 ∆(H).
Sarah C Rundell, Jane H Long. The Hodge Structure of the Coloring Complex of a Hypergraph (Extended Abstract). 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), 2010, San Francisco, United States. pp.1017-1024. hal-01186258
Tell us how this article helped you.