We investigate numerically and analytically 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 thin random graphs in this limit are locally tree-like, in distinction to the 'fat' random graphs that appear in the planar Feynman diagram limit, N ? 8, more familiar from discretized models of two-dimensional gravity. The interest of the thin graphs is that they give mean-field theory behaviour for spin models living on them without infinite range interactions or the boundary problems of genuine tree-like structures such as the Bethe lattice. q-state Potts models display a first-order transition in the mean field for q > 2, so the thin-graph Potts models provide a useful test case for exploring discontinuous transitions in mean-field theories in which many quantities can be calculated explicitly in the saddle-point approximation. Such discontinuous transitions also appear in multiple Ising models on thin graphs and may have implications for the use of the replica trick in spin-glass models on random graphs.