The diameter of randomly twisted hypercubes

TL;DR

This paper improves the upper bound on the diameter of the random twisted hypercube, showing it is asymptotically close to n divided by log base 2 of n, with high probability.

Contribution

The authors establish a tighter asymptotic bound on the diameter of the random twisted hypercube, refining previous results.

Findings

Diameter is asymptotically (1 + o(1)) n / log_2 n with high probability.

Previous upper bound was O(n log log log n / log log n).

Diameter lower bound is (n - 1) / log_2 n.

Abstract

The $n$-dimensional random twisted hypercube $\mathbf{G}_n$ is constructed recursively by taking two instances of $\mathbf{G}_{n-1}$, with any joint distribution, and adding a random perfect matching between their vertex sets. Benjamini, Dikstein, Gross, and Zhukovskii showed that its diameter is $O(n\log \log \log n/\log \log n)$ with high probability and at least ${(n - 1)/ \log_2 n}$. We improve their upper bound by showing that $$\operatorname{diam}(\mathbf{G}_n) = \big(1 + o(1)\big) \frac{n}{\log_2 n}$$ with high probability.