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Statement

Setting (p. 1). For $d\ge2$, $c(n,d)$ is the largest number of edges of the contact graph of a packing of $n$ non-overlapping translates of the unit ball $\mathbf B^d$ in $\mathbb E^d$, that is, the largest number of touching pairs among $n$ non-overlapping unit balls.

Theorem 3.1 (p. 3). For every $n\ge2$, $c(n,2)=\lfloor 3n-\sqrt{12n-3}\rfloor$.

The survey attributes the theorem to Harborth (its reference [26], Elem. Math. 29 (1974), 14--15) and adds (p. 3) that the hexagonal arrangement, built from a disk surrounded by six disks and continued in hexagonal layers, attains $c(n,2)$ for every $n$. It records the consequence (1) (p. 4): $\lim_{n\to+\infty}(3n-c(n,2))/\sqrt n=\sqrt{12}$.

Proof pointer

Not proved in the survey; the proof is Harborth's, in the cited note.

Read depth

Claims checked: the definition of $c(n,d)$ and Theorem 3.1 were read on the page images of the print. The cited proof was not read. Nothing here is independently reviewed.

Dependencies

None in the corpus. External input: Harborth's note, which the survey cites for the proof.

Source. K. Bezdek and M. A. Khan, Contact numbers for sphere packings, in New Trends in Intuitive Geometry, Bolyai Society Mathematical Studies, Springer (2018), 25--47, doi:10.1007/978-3-662-57413-3_2; the label and page are those of arXiv:1601.00145v2, the edition read, named on the source card.

Bears on

• Problem 1084: centers of touching unit disks are at distance $2$ and centers of non-overlapping ones at distance at least $2$, so after scaling by $1/2$ the theorem states $f_2(n)=\lfloor 3n-\sqrt{12n-3}\rfloor$ for every $n\ge2$, the planar case of the problem. The survey restates the result; the problem's claim page credits Harborth's note.