# Contact graphs of unit sphere packings revisited

@article{Bezdek2012ContactGO,
title={Contact graphs of unit sphere packings revisited},
author={K{\'a}roly Bezdek and Samuel Reid},
journal={Journal of Geometry},
year={2012},
volume={104},
pages={57-83}
}
• Published 21 October 2012
• Mathematics
• Journal of Geometry
The contact graph of an arbitrary finite packing of unit balls in Euclidean 3-space is the (simple) graph whose vertices correspond to the packing elements and whose two vertices are connected by an edge if the corresponding two packing elements touch each other. One of the most basic questions on contact graphs is to find the maximum number of edges that a contact graph of a packing of n unit balls can have. In this paper, improving earlier estimates, we prove that the number of touching pairs…
27 Citations

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## References

SHOWING 1-10 OF 21 REFERENCES
Contact Numbers for Congruent Sphere Packings in Euclidean 3-Space
• K. Bezdek
• Mathematics, Computer Science
Discret. Comput. Geom.
• 2012
The contact graph of an arbitrary finite packing of unit balls in Euclidean 3-space is the (simple) graph whose vertices correspond to the packing elements and whose two vertices are connected by an
Representing graphs by disks and balls (a survey of recognition-complexity results)
• Computer Science, Mathematics
Discret. Math.
• 2001
It is proved that the recognition of disk-intersection graphs (in the unbounded ratio case) is NP-hard, and some partial results concerning recognition of ball intersection and contact graphs in higher dimensions are shown.
On the Maximum Number of Touching Pairs in a Finite Packing of Translates of a Convex Body
• K. Bezdek
• Computer Science, Mathematics
J. Comb. Theory, Ser. A
• 2002
Upper bounds for the maximum number of touching pairs in a packing of n translates of a given convex body in Ed, d?3 are given.
Dense Sphere Packings: A Blueprint for Formal Proofs
The 400-year-old Kepler conjecture asserts that no packing of congruent balls in three dimensions can have a density exceeding the familiar pyramid-shaped cannonball arrangement. In this book, a new
Lectures on discrete geometry
• J. Matousek
• Computer Science, Mathematics
• 2002
This book is primarily a textbook introduction to various areas of discrete geometry, in which several key results and methods are explained, in an accessible and concrete manner, in each area.
The isoperimetric inequality
where A is the area enclosed by a curve C of length L, and where equality holds if and only if C is a circle. The purpose of this paper is to recount some of the most interesting of the many
A generalization of the Discrete Isoperimetric Inequality for Piecewise Smooth Curves of Constant Geodesic Curvature
• Mathematics, Computer Science
Period. Math. Hung.
• 2006
The discrete isoperimetric inequality for polygons'' on the three planes of constant curvature bounded by arcs of a given constant geodesic curvature is extended.
Low-dimensional lattices. VI. Voronoi reduction of three-dimensional lattices
• Mathematics
Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences
• 1992
The aim of this paper is to describe how the Voronoi cell of a lattice changes as that lattice is continuously varied. The usual treatment is simplified by the introduction of new parameters called
Foundations of Hyperbolic Manifolds
This book is an exposition of the theoretical foundations of hyperbolic manifolds. It is intended to be used both as a textbook and as a reference. The reader is assumed to have a basic knowledge of
Packing and Covering
• G. Tóth
• Mathematics, Computer Science
Handbook of Discrete and Computational Geometry, 2nd Ed.
• 2004
In this age of modern era, the use of internet must be maximized to get the on-line packing and covering book, as the world window, as many people suggest.