On contact graphs of totally separable domains

Abstract

Contact graphs have emerged as an important tool in the study of translative packings of convex bodies. The contact number of a packing of translates of a convex body is the number of edges in the contact graph of the packing, while the Hadwiger number of a convex body is the maximum vertex degree over all such contact graphs. In this paper, we investigate the Hadwiger and contact numbers of totally separable packings of two-dimensional convex bodies, which we refer to as the separable Hadwiger number and the separable contact number, respectively. We show that the separable Hadwiger number of any two-dimensional smooth convex body is 4 and the maximum separable contact number of any packing of n translates of a two-dimensional smooth strictly convex body is $$\left\lfloor 2n - 2\sqrt{n}\right\rfloor $$ . Our proofs employ a characterization of total separability in terms of hemispherical caps on the boundary of a convex body, Auerbach bases of finite dimensional real normed spaces, angle measures in real normed planes, minimal perimeter polyominoes and an approximation of two-dimensional $${{\mathrm{\mathbf {o}}}}$$ -symmetric smooth strictly convex bodies by, what we call, Auerbach domains.