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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A Hodge decomposition interpretation for the coefficients of the chromatic polynomial
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by Phil Hanlon PDF
Proc. Amer. Math. Soc. 136 (2008), 3741-3749 Request permission

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
  • Email: hanlon@umich.edu
  • 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
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 136 (2008), 3741-3749
  • MSC (2000): Primary 05C15; Secondary 18G35
  • DOI: https://doi.org/10.1090/S0002-9939-08-08974-0
  • MathSciNet review: 2425711