Dichromatic polynomials and Potts models summed over rooted maps

We consider the sum of dichromatic polynomials over non-separable rooted planar maps, an interesting special case of which is the enumeration of such maps. We present some known results and derive new ones. The general problem is equivalent to the $q$-state Potts model randomized over such maps. Like the regular ferromagnetic lattice models, it has a first-order transition when $q$ is greater than a critical value $q_c$, but $q_c$ is much larger - about 72 instead of 4.