Nerve graphs of Euclidean sphere packings

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For a positive integer $d$, a sphere packing in $\mathbb R^d$ is a family of closed Euclidean balls $B(c_v,r_v)$ with radii $r_v>0$ and pairwise disjoint interiors. Its tangency graph has one vertex $v$ per ball and an edge $\{v,w\}$ exactly when $\|c_v-c_w\|_2=r_v+r_w$.

Statement recorded in the supplied source.

For every $d$, characterize the graphs that are tangency graphs of such sphere packings in $\mathbb R^d$. [S2]

Short English statement

Which graphs can be realized by touching Euclidean balls with disjoint interiors in each dimension?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10000063 (AMR-099-0063). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Itai Benjamini, Euclidean vs. Graph Metric, version dated 16 July 2012; original author manuscript preserved in Arquivo.pt capture of 31 December 2020, Section 6, Question 6.1 and subsequent definition, p. 7. https://arquivo.pt/noFrame/replay/20201231041538id_/http://www.wisdom.weizmann.ac.il/~itai/erd100.pdf.