Dichromatic Polynomials and Potts Models Summed Over Rooted Maps

Abstract.

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: it remains an open question whether this model exhibits a phase transition or critical behaviour.