The coloring complex and cyclic coloring complex of a complete k-uniform hypergraph

TL;DR

This paper investigates the homology of coloring and cyclic coloring complexes of complete k-uniform hypergraphs, revealing shellability, homology ranks, and connections to chromatic polynomials and algebraic structures.

Contribution

It establishes shellability of the coloring complex, determines homology ranks in terms of chromatic polynomials, and links homology dimensions to binomial coefficients and algebraic structures.

Findings

Coloring complex is shellable.

Homology rank relates to chromatic polynomial.

Homology dimensions given by binomial coefficients.

Abstract

In this paper, we study the homology of the coloring complex and the cyclic coloring complex of a complete $k$-uniform hypergraph. We show that the coloring complex of a complete $k$-uniform hypergraph is shellable, and we determine the rank of its unique nontrivial homology group in terms of its chromatic polynomial. We also show that the dimension of the $(n-k-1)^{st}$ homology group of the cyclic coloring complex of a complete $k$-uniform hypergraph is given by a binomial coefficient. Further, we discuss a complex whose $r$-faces consist of all ordered set partitions $[B_1, \hdots, B_{r+2}]$ 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 $\C[x_1, \hdots, x_n]/ \{x_{i_1} \hdots x_{i_k} \mid i_{1} \hdots i_{k}$ is a hyperedge of $H \}$.