Phase diagrams of the q-state potts model on Bethe lattices

We use a rigorous, recently developed, method of solution of spin models on Bethe lattices (Cayley trees). It allows us to detect all the phases of the ferromagnetic and antiferromagnetic q-state Potts models in a field (FPM and APM respectively). The phase diagrams of the FPM are characterized by three critical lines of first-order transitions. In the APM the ordered phases (which appear always for h > 0, and under certain conditions for h= 0 and/or h<0) are separated from the disordered one by phase boundaries characterized by second-order transitions, and are separated one from the other by critical lines characterized by first-order and second-order transitions. Several results are found, which hold true on standard lattices, too.