  Recent results establish for 2-spin antiferromagnetic systems that the
computational complexity of approximating the partition function on graphs of
maximum degree D undergoes a phase transition that coincides with the
uniqueness phase transition on the infinite D-regular tree. For the
ferromagnetic Potts model we investigate whether analogous hardness results
hold. Goldberg and Jerrum showed that approximating the partition function of
the ferromagnetic Potts model is at least as hard as approximating the number
of independent sets in bipartite graphs (#BIS-hardness). We improve this
hardness result by establishing it for bipartite graphs of maximum degree D. We
first present a detailed picture for the phase diagram for the infinite
D-regular tree, giving a refined picture of its first-order phase transition
and establishing the critical temperature for the coexistence of the disordered
and ordered phases. We then prove for all temperatures below this critical
temperature that it is #BIS-hard to approximate the partition function on
bipartite graphs of maximum degree D. As a corollary, it is #BIS-hard to
approximate the number of k-colorings on bipartite graphs of maximum degree D
when k <= D/(2 ln D).
  The #BIS-hardness result for the ferromagnetic Potts model uses random
bipartite regular graphs as a gadget in the reduction. The analysis of these
random graphs relies on recent connections between the maxima of the
expectation of their partition function, attractive fixpoints of the associated
tree recursions, and induced matrix norms. We extend these connections to
random regular graphs for all ferromagnetic models and establish the Bethe
prediction for every ferromagnetic spin system on random regular graphs. We
also prove for the ferromagnetic Potts model that the Swendsen-Wang algorithm
is torpidly mixing on random D-regular graphs at the critical temperature for
large q.
