Percolation on a Feynman Diagram

In a recent paper hep-lat/9704020 we investigated Potts models on ``thin'' random graphs -- generic Feynman diagrams, using the idea that such models may be expressed as the N --> 1 limit of a matrix model. The models displayed first order transitions for all q greater than 2, giving identical behaviour to the corresponding Bethe lattice. We use here one of the results of hep-lat/9704020 namely a general saddle point solution for a q state Potts model expressed as a function of q, to investigate some peculiar features of the percolative limit q -> 1 and compare the results with those on the Bethe lattice.