The string of diamonds is nearly tight for rumour spreading

For a rumour spreading protocol, the spread time is defined as the first time that everyone learns the rumour. We compare the synchronous push&pull rumour spreading protocol with its asynchronous variant, and show that for any $n$-vertex graph and any starting vertex, the ratio between their expected spread times is bounded by $O\left(n^{1\mathrm{/}3}{\log }^{2\mathrm{/}3}n\right)$. This improves the $O(\sqrt{n})$ upper bound of Giakkoupis, Nazari, and Woelfel (in Proceedings of ACM Symposium on Principles of Distributed Computing, 2016). Our bound is tight up to a factor of $O(\log n)$, as illustrated by the string of diamonds graph. We also show that if for a pair $\alpha ,\beta$ of real numbers, there exists infinitely many graphs for which the two spread times are $n^{\alpha }$ and $n^{\beta }$ in expectation, then $0\le \alpha \le 1$ and $\alpha \le \beta \le \frac{1}{3}+\frac{2}{3}\alpha$; and we show each such pair $\alpha ,\beta$ is achievable.