The diameter of randomly twisted hypercubes

Abstract

The $n$-dimensional random twisted hypercube $G_n$ is constructed recursively by taking two instances of $G_{n-1}$, with any joint distribution, and adding a random perfect matching between their vertex sets. Benjamini, Dikstein, Gross, and Zhukovskii showed that its diameter is $O(nlogloglogn/loglogn)$ with high probability and at least $(n-1)/{log}_{2}n$. We improve their upper bound by showing that $\mathrm{diam}\left(G_n\right)=\left(1+o\left(1\right)\right)\frac{n}{{log}_{2}n}$ with high probability.