On contact graphs of totally separable domains

TL;DR

This paper studies the maximum contact and Hadwiger numbers in totally separable packings of convex domains, establishing exact values for smooth convex bodies and bounds for general cases using geometric and normed space techniques.

Contribution

It introduces the concepts of separable Hadwiger and contact numbers and provides exact and asymptotic bounds for these in totally separable packings of convex domains.

Findings

Separable Hadwiger number of any smooth convex domain is 4.

Maximum separable contact number of n translates of a smooth strictly convex domain is approximately 2n - 2√n.

Uses geometric characterizations and normed space methods to derive results.

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 convex domains, which we refer to as the separable Hadwiger number and the separable contact number, respectively. We show that the separable Hadwiger number of any smooth convex domain is $4$ and the maximum separable contact number of any packing of $n$ translates of a smooth strictly convex domain 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 smooth $o$-symmetric strictly convex domains by, what we call, Auerbach domains.