  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
Blume-Emery-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)=[4 z / 3(z-1)]
[(z-1)/(z-2)]^z invisible states are required to manifest the equivalent
Blume-Emery-Griffiths tricriticality. When z=3, the 3-regular, random-graph
result is recovered, while the infinite z limit delivers the Bragg-Williams,
mean-field result.
