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Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs
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Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs
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Journal Article
Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs
2012
Overview
The chromatic polynomial χG(q)\\chi _G(q) of a graph GG counts the number of proper colorings of GG. 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 χG(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 Pn×Pm\\mathbb {P}^n \\times \\mathbb {P}^m corresponding to subvarieties and answer a question posed by Trung-Verma.
Publisher
American Mathematical Society
Subject
