Disproof of a conjecture by Erdős and Guy on the crossing number of hypercubes
Yuansheng Yang, Guoqing Wang, Haoli Wang, Yan Zhou · arXiv (Cornell University) · 2012
Let $Q_n$ be the $n$-dimensional hypercube, and let ${\rm cr}(Q_n)$ be the \textit{crossing number} of $Q_n$. Erdős 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.