On the Roots of σ-Polynomials

Given a graph G of order n, the sigma-polynomial of G is the generating function sigma(G,x)=Sigma a(i)x(i) where ai is the number of partitions of the vertex set of G into i nonempty independent sets. Such polynomials arise in a natural way from chromatic polynomials. Brenti (Trans Am Math Soc 332 (1992), 729-756) proved that sigma-polynomials of graphs with chromatic number at least n-2 had all real roots, and conjectured the same held for chromatic number n-3. We affirm this conjecture.