Abstract.The number of so-called invisible states which need to be added to the q-state Potts model to transmute its phase transition from continuous to first order has attracted recent attention. In the q = 2 case, a Bragg-Williams, mean-field approach necessitates four such invisible states while a 3-regular, random-graph formalism requires seventeen. In both of these cases, the changeover from second-to first-order behaviour induced by the invisible states is identified through the tricritical point of an equivalent BlumeEmery-Griffiths model.Here we investigate the generalised Potts model on a Bethe lattice with z neighbours. We show that, in the q = 2 case, r c (z) = 4z 3(z − 1)states are required to manifest the equivalent Blume-Emery-Griffiths tricriticality. When z = 3, the 3-regular, random-graph result is recovered, while z → ∞ delivers the Bragg-Williams, mean-field result.arXiv:1307.2803v1 [cond-mat.stat-mech]