Pure Maths Colloquium: Sean Dewar

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 \(m > 1\) vertices has at most \(2m-3\) edges, \(G\) can be realised as the contact graph of a random disc packing with non-zero probability. I will discuss how these ideas can be extended to homothetic packings of a centrally symmetric (c.s.) convex bodies. The main results are (i) for any strictly convex and smooth c.s. convex body \(\mathcal{C}\), every random homothetic packing of \(\mathcal{C}\) 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 \(\mathcal{C}\), we can realise all \((2,2)\)-sparse planar graphs as the contact graph of a random homothetic packing with positive probability.