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

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:119

Article

Rundell, Sarah Crown

关键词: Coloring complex;    Cyclic coloring complex;    Complete k-uniform hypergraph;   

DOI  :  10.1016/j.jcta.2012.02.001

来源: Elsevier

【 摘 要 】

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,..., Br+2] where none of the B-i contain a hyperedge of the complete k-uniform hypergraph H and where 1 is an element of 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),...., x(n)]/{X-i1 ... X-ik vertical bar i(1) ... i(k) is a hyperedge of H}. (C) 2012 Elsevier Inc. All rights reserved.

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