Density of Chromatic Roots in Minor-Closed Graph Families

We prove that the roots of the chromatic polynomials of planar graphs are dense in the interval between 32/27 and 4, except possibly in a small interval around Ï„ + 2 where Ï„ is the golden ratio. This interval arises due to a classical result of Tutte, which states that the chromatic polynomial of every planar graph takes a positive value at Ï„ + 2. Our results lead us to conjecture that Ï„ + 2 is the only such number less than 4.