Noninteger Beraha numbers can never be roots of any chromatic polynomials with the possible exception of (G. Royle, pers. comm., Nov. 21, 2005). However,
the roots of chromatic polynomials of planar triangulations appear to cluster around
the Beraha numbers (and, technically, are conjectured to be accumulation points of
roots of planar triangulation chromatic polynomials).