I. General Theory and Square-Lattice Chromatic Polynomial

We study the chromatic polynomials (= zero-temperature antiferromagnetic Potts-model partition functions) PG(q) for m × n rectangular subsets of the square lattice, with m ≤ 8 (free or periodic transverse boundary conditions) and n arbitrary (free longitudinal boundary conditions), using a transfer matrix in the Fortuin-Kasteleyn representation. In particular, we extract the limiting curves of partition-function zeros when n → ∞, which arise from the crossing in modulus of dominant eigenvalues (Beraha–Kahane–Weiss theorem). We also provide evidence that the Beraha numbers B2, B3, B4, B5 are limiting points of partition-function zeros as n → ∞ whenever the strip width m is ≥ 7 (periodic transverse b.c.) or ≥ 8 (free transverse b.c.). Along the way, we prove that a noninteger Beraha number (except perhaps B10) cannot be a chromatic root of any graph.