  We provide a new criterion based on graph duality to predict whether the
3-state Potts antiferromagnet on a plane quadrangulation has a zero- or
finite-temperature critical point, and its universality class. The former case
occurs for quadrangulations of self-dual type, and the zero-temperature
critical point has central charge $c=1$. The latter case occurs for
quadrangulations of non-self-dual type, and the critical point belongs to the
universality class of the 3-state Potts ferromagnet. We have tested this
criterion against high-precision computations on four lattices of each type,
with very good agreement. We have also found that the Wang-Swendsen-Koteck\'y
algorithm has no critical slowing-down in the former case, and critical
slowing-down in the latter.
