  We address the enumeration of properly q-colored planar maps, or more
precisely, the enumeration of rooted planar maps M weighted by their chromatic
polynomial \chi_M(q) and counted by the number of vertices and faces. We prove
that the associated generating function is algebraic when q\not=0,4 is of the
form 2+2 cos (j\pi/m), for integers j and m. This includes the two integer
values q=2 and q=3. We extend this to planar maps weighted by their Potts
polynomial P_M(q,\nu), which counts all q-colorings (proper or not) by the
number of monochromatic edges. We then prove similar results for planar
triangulations, thus generalizing some results of Tutte which dealt with their
proper q-colorings. In statistical physics terms, the problem we study consists
in solving the Potts model on random planar lattices. From a technical
viewpoint, this means solving non-linear equations with two "catalytic"
variables. To our knowledge, this is the first time such equations are being
solved since Tutte's remarkable solution of properly q-colored triangulations.
