Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs

Huh, June^1, 2

¹ Department of Mathematics, University of Illinois, Urbana, Illinois 61801
² Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109

Journal of the American Mathematical Society, Tome 25 (2012) no. 3, pp. 907-927

Résumé

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.

Détail

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

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