کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
6423592 | 1342425 | 2011 | 10 صفحه PDF | دانلود رایگان |
Let G be a simple graph with n vertices. The coloring complex Î(G) was defined by SteingrÃmsson, and the homology of Î(G) was shown to be nonzero only in dimension nâ3 by Jonsson. Hanlon recently showed that the Eulerian idempotents provide a decomposition of the homology group Hnâ3(Î(G)) where the dimension of the jth component in the decomposition, Hnâ3(j)(Î(G)), equals the absolute value of the coefficient of λj in the chromatic polynomial of G, ÏG(λ).Let H be a hypergraph with n vertices. In this paper, we define the coloring complex of a hypergraph, Î(H), and show that the coefficient of λj in ÏH(λ) gives the Euler Characteristic of the jth Hodge subcomplex of the Hodge decomposition of Î(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 ÏH(λ) equals the dimension of the jth Hodge piece of the Hodge decomposition of Î(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.
⺠We define the coloring complex of a hypergraph. ⺠A chromatic polynomial is related to the Euler Characteristic of the Hodge subcomplexes. ⺠We examine when the Hodge subcomplexes of the coloring complex are Cohen-Macaulay.
Journal: Discrete Mathematics - Volume 311, Issue 20, 28 October 2011, Pages 2164-2173