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A Hodge decomposition interpretation for the coefficients of the chromatic polynomial

Author: Phil Hanlon
Journal: Proc. Amer. Math. Soc. 136 (2008), 3741-3749
MSC (2000): Primary 05C15; Secondary 18G35
Published electronically: June 17, 2008
MathSciNet review: 2425711
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Abstract: Let $ G$ be a simple graph with $ n$ nodes. The coloring complex of $ G$, as defined by Steingrimsson, has $ r$-faces consisting of all ordered set partitions, $ (B_1, \ldots ,B_{r+2})$ in which at least one $ B_i$ contains an edge of $ G$. 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 $ G$.

In this paper, we show that the Eulerian idempotents give a decomposition of the top homology of $ G$ into $ n-1$ 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 $ G$. 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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Additional Information

Phil Hanlon
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1109

Received by editor(s): January 24, 2006
Received by editor(s) in revised form: August 11, 2006, and October 16, 2006
Published electronically: June 17, 2008
Communicated by: John R. Stembridge
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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