The Crossing Number of the Hypercube

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.

The hypercube $Q_d$ has vertex set $\{0,1\}^d$, with two vertices adjacent when they differ in exactly one coordinate. The crossing number $\operatorname{cr}(G)$ is the minimum number of interior edge crossings over plane drawings of $G$; drawings may be taken in general position, with no triple crossings and no edge passing through a nonincident vertex.

Statement recorded in the supplied source.

Does the asymptotic identity\[ \lim_{d\to\infty}\frac{\operatorname{cr}(Q_d)}{4^d}=\frac5{32} \] hold? [S2]

Short English statement

Is five thirty-seconds the leading constant for the crossing number of the high-dimensional hypercube, normalized by four to the dimension?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3320 (OPG-313). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, The Crossing Number of the Hypercube, node 313, original problem page. Original question and full discussion. https://www.openproblemgarden.org/op/the_crossing_number_of_the_hypercube.