Broken-Cycle-Free Subgraphs and the Log-Concavity Conjecture for Chromatic Polynomials

In a paper from 1912 aimed at proving the four-color theorem, G. D. Birkhoff [Birkhoff 12] introduced a function P (G, x), defined for all positive integers x to be the number of proper x-colorings of the graph G. As it turns out, P (G, x) is a polynomial in x and so is defined for all real and complex values of x as well. Of course, P (G, x) is the by now well-known chromatic polynomial, and although Birkhoff’s original hope that it would help resolve the four-color conjecture did not bear fruit, it has attracted a steady stream of attention through the years. Most of the investigations regarding the chromatic polynomial have focused on the location of its zeros. An early example is the work of Tutte on the chromatic roots of triangulations and the so-called golden identity, nicely described in [Tutte 98]. More recently we have the results of Thomassen on zero-free intervals of minor closed graph families [Thomassen 97] and the influence of Hamiltonian paths on the zeros of the chromatic polynomial [Thomassen 00]. There has also been a recent influx of ideas from statistical physics due to the connection to the Potts model. Using this connection, Sokal [Sokal 01] has shown that the moduli of the zeros