The number of touching pairs of congruent sphere packings in Euclidean 3-space

A packing of $n$ congruent balls in $\mathbb{R}^3$ is a family of interior-disjoint Euclidean balls all having the same radius. The contact number of a packing is the number of touching pairs of balls. In this paper we investigate the problem of determining the maximum contact number, $c(n)$, of a packing of $n$ congruent balls in $\mathbb{R}^3$. We first show that all packings of $n$ congruent balls that have a contact number of $c(n)$ are minimally rigid. Furthermore, we show that $c(n)=3n-6$ for $n=6,7,8,$ and $9$. These two results resolve a conjecture of K. Bezdek and Khan. During the proof of the latter result, we also enumerate the contact structures of all packings of $n$ congruent balls with contact number $c(n)$ for $n=6,7,$ and $8$. Additionally, we provide a lower bound construction which shows $c(n)> 6n-6\sqrt[3]{2}n^\frac{2}{3}$ when $n=16k^3-33k^2+24k-6$ where $k\in \mathbb{N}$. We also look at the restricted problem where each ball is centered on the face-centered cubic lattice $A_3$. In this case let $c_{A}(n)$ denote the maximum contact number. We show that $c_{A}(n)\leq 6n-\frac{6}{\sqrt[6]{2}}n^\frac{2}{3}$ for all $n$, and determine the asymptotics of $c_{A}(n)$ to be $c_{A}(n)=6n-(1+o(1))6\sqrt[3]{2}n^\frac{2}{3}$.