  In some recent papers by Tamura, Tanaka and Kawashima [arXiv:1102.5475,
arXiv:1012.4254], a class of Potts models with "invisible" states was
introduced, for which the authors argued by numerical arguments and by a
mean-field analysis that a first-order transition occurs. Here we show that the
existence of this first-order transition can be proven rigorously, by
relatively minor adaptations of existing proofs for ordinary Potts models. In
our argument we present a random-cluster representation for the model, which
might be of independent interest.
