Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs

Huh, June

doi:10.1090/S0894-0347-2012-00731-0

Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs
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by June Huh;

J. Amer. Math. Soc. 25 (2012), 907-927

DOI: https://doi.org/10.1090/S0894-0347-2012-00731-0

Published electronically: February 8, 2012

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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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