On contact graphs of totally separable packings in low dimensions

TL;DR

This paper investigates the properties of contact graphs in totally separable packings of convex bodies in low-dimensional Euclidean spaces, providing bounds on contact graph complexity using geometric and algebraic methods.

Contribution

It establishes upper bounds on the maximum vertex degree and edge count of contact graphs for totally separable packings of smooth convex bodies in dimensions 2 to 4.

Findings

Bound on maximum vertex degree (separable Hadwiger number)

Bound on maximum number of edges (maximum separable contact number)

Application of convexity, linear algebra, and volumetric estimates

Abstract

The contact graph of a packing of translates of a convex body in Euclidean $d$-space $\mathbb E^d$ is the simple graph whose vertices are the members of the packing, and whose two vertices are connected by an edge if the two members touch each other. A packing of translates of a convex body is called totally separable, if any two members can be separated by a hyperplane in $\mathbb E^d$ disjoint from the interior of every packing element. We give upper bounds on the maximum vertex degree (called separable Hadwiger number) and the maximum number of edges (called maximum separable contact number) of the contact graph of a totally separable packing of $n$ translates of an arbitrary smooth convex body in $\mathbb E^d$ with $d=2,3,4$. In the proofs, linear algebraic and convexity methods are combined with volumetric and packing density estimates based on the underlying isoperimetric (resp., reverse isoperimetric) inequality.