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

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

2011-10-22

arxiv.org/abs/1110.5007v1

Mathematics

Combinatorics

arXiv admin note: text overlap with arXiv:1105.4820

Scientific paper

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 $H \}$.

Affiliated with

Also associated with

No associations

Rating

The coloring complex and cyclic coloring complex of a complete k-uniform hypergraph does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with The coloring complex and cyclic coloring complex of a complete k-uniform hypergraph, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The coloring complex and cyclic coloring complex of a complete k-uniform hypergraph will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-526415