Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs

TL;DR

This paper proves the log-concavity of chromatic polynomial coefficients, introduces Milnor number analogues for projective hypersurfaces, and links these concepts to graph coloring and algebraic geometry.

Contribution

It confirms the log-concavity conjecture for chromatic polynomial coefficients and establishes a new connection between algebraic invariants and graph theory.

Findings

Proved the log-concavity of chromatic polynomial coefficients.

Defined Milnor number analogues for projective hypersurfaces.

Established a relation between Milnor numbers and Newton polytopes.

Abstract

The chromatic polynomial 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. We define a sequence of numerical invariants of projective hypersurfaces analogous to the Milnor number of local analytic hypersurfaces. Then we give a characterization of correspondences between projective spaces up to a positive integer multiple which includes the conjecture on the chromatic polynomial as a special case. As a byproduct of our approach, we obtain an analogue of Kouchnirenko's theorem relating the Milnor number with the Newton polytope.