Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs

Source abstract

The chromatic polynomial χ G ( q ) \chi _G(q) of a graph G G counts the number of proper colorings of G 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 χ G ( q ) \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 P n × P m \mathbb {P}^n \times \mathbb {P}^m corresponding to subvarieties and answer a question posed by Trung-Verma.