Given n discs with randomly selected radii, it was shown by Connelly, Gortler and Theran that there can be at most 2n-3 contacts in any packing involving the n discs. They also conjectured that the opposite is true; given a planar graph G where every subgraph on n' vertices has at most 2n'-3 edges, G can be realised as the contact graph of a random disc packing with non-zero probability. These ideas can be extended to packings of a much larger class of convex body; namely those that are centrally symmetric (c.s.), strictly convex and smooth. In the presentation we shall discuss the following two results: (i) for any suitable c.s. convex body C, every random homothetic packing of C has a (2,2)-sparse contact graph (i.e. all subgraphs of G on n vertices have at most 2n-2 edges); (ii) for almost every c.s. convex body C, we can realise all (2,2)-sparse planar graphs as the contact graph of a random homothetic packing with non-zero probability.