A Hodge decomposition interpretation for the coefficients of the chromatic polynomial期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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A Hodge decomposition interpretation for the coefficients of the chromatic polynomial

Authors:	Phil Hanlon

Institution:	Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1109

Abstract:	Let be a simple graph with nodes. The coloring complex of , as defined by Steingrimsson, has -faces consisting of all ordered set partitions, $(B_1, \ldots ,B_{r+2})$ in which at least one contains an edge of . Jonsson proved that the homology $H_{*}(G)$ of the coloring complex is concentrated in the top degree. In addition, Jonsson showed that the dimension of the top homology is one less than the number of acyclic orientations of . In this paper, we show that the Eulerian idempotents give a decomposition of the top homology of into components $H_{n-3}^{(j)}(G)$ . We go on to prove that the dimensions of the Hodge pieces of the homology are equal to the absolute values of the coefficients of the chromatic polynomial of . Specifically, if we write $\chi_G(\lambda) = (\sum_{j=1}^{n-1} c_j (-1)^{n-j} \lambda^j) + \lambda^n$ , then $dim(H_{n-3}^{(j)}(G)) = c_j$ .

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