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Statement

Setting (pp. 5--6). In Section 5 a sphere is a unit sphere in $\mathbb E^3$ and a finite packing of them is a cluster. By Definition 1 (p. 6), a cluster of $n\ge4$ unit spheres is minimally rigid when each sphere touches at least $3$ others and the cluster has at least $3n-6$ contacts.

Proposition 5.1 (p. 6). Assume that every maximal contact minimally rigid packing of $n\le9$ spheres is listed in the work of Arkus, Manoharan and Brenner (the survey's references [3], arXiv:1011.5412v2, and [4], SIAM J. Discrete Math. 25 (2011)). Then for $n=4,\ldots,9$,

$$c(n,3)=3n-6,$$

and some minimally rigid cluster has $c(n,3)$ contacts.

The hypothesis is not proved: the survey says (p. 6) that the list for $n\le9$ is putatively complete up to possible omissions due to round-off errors, and (p. 7) that the list could potentially be incomplete.

Proof pointer

Pp. 6--7. The proof shows by induction on $n$, from $n=4$, that every contact-maximal graph on $4\le n\le9$ vertices has minimum degree at least $3$ and an exposed triangle: three mutually touching spheres to which a further sphere can be attached without overlap. A vertex of degree $2$ could be deleted and reattached to an exposed triangle of a contact-maximal graph on the remaining vertices, giving more contacts. The claim for these $n$ is checked exhaustively on the assumed list.

Read depth

Claims checked: Definition 1, the proposition and its proof were read on the page images of the print. The computer-generated lists the hypothesis concerns were not examined. Nothing here is independently reviewed.

Dependencies

None in the corpus. External input: the enumeration of minimally rigid packings by Arkus, Manoharan and Brenner, assumed complete.

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: after scaling by $1/2$, under the stated completeness assumption it gives $f_3(n)=3n-6$ for $n=4,\ldots,9$. It is conditional on a computer enumeration being complete.