Disproof of a conjecture by Erd\H{o}s and Guy on the crossing number of hypercubes

TL;DR

This paper disproves a long-standing conjecture by Erdős and Guy regarding the crossing number of hypercubes by providing a new drawing with fewer crossings for dimensions greater than six.

Contribution

It introduces a novel construction of hypercube drawings that reduces crossings, thereby disproving the previously believed exact formula for all dimensions.

Findings

For n > 6, the crossing number is less than the conjectured value.

A new drawing method achieves fewer crossings than the conjecture predicted.

The conjecture by Erdős and Guy is false for hypercubes of dimension greater than six.

Abstract

Let $Q_n$ be the $n$-dimensional hypercube, and let ${\rm cr}(Q_n)$ be the \textit{crossing number} of $Q_n$. Erd\H{o}s and Guy in 1973 conjectured the following equality: ${\rm cr}(Q_n)=\frac{5}{32}4^n-\lfloor\frac{n^2+1}{2}\rfloor 2^{n-2}$. In this paper, we construct a drawing of $Q_n$ with less crossings when $n>6$, which implies that for $n>6$ we have a strict inequality.