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Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs

Author: June Huh
Journal: J. Amer. Math. Soc. 25 (2012), 907-927
MSC (2010): Primary 14B05, 05B35
Published electronically: February 8, 2012
MathSciNet review: 2904577
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Abstract: The chromatic polynomial $\chi _G(q)$ of a graph $G$ counts the number of proper colorings of $G$. We give an affirmative answer to the conjecture of Read and Rota-Heron-Welsh that the absolute values of the coefficients of the chromatic polynomial form a log-concave sequence. The proof is obtained by identifying $\chi _G(q)$ with a sequence of numerical invariants of a projective hypersurface analogous to the Milnor number of a local analytic hypersurface. As a by-product of our approach, we obtain an analogue of Kouchnirenko’s theorem relating the Milnor number with the Newton polytope; we also characterize homology classes of $\mathbb {P}^n \times \mathbb {P}^m$ corresponding to subvarieties and answer a question posed by Trung-Verma.

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Additional Information

June Huh
Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801
Address at time of publication: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
MR Author ID: 974745

Keywords: Chern-Schwartz-MacPherson class, characteristic polynomial, chromatic polynomial, Milnor number, Okounkov body
Received by editor(s): July 10, 2011
Received by editor(s) in revised form: January 17, 2012
Published electronically: February 8, 2012
Additional Notes: The author acknowledges support from National Science Foundation grant DMS 0838434 “EMSW21-MCTP: Research Experience for Graduate Students”.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.