The thresholds for diameter 2 in random Cayley graphs

TL;DR

This paper determines the threshold probabilities for random Cayley graphs of groups to have diameter at most 2, showing these thresholds are tight and highlighting cases where Cayley graphs reach diameter 2 faster than Erdős-Rényi graphs.

Contribution

It establishes precise probabilistic thresholds for diameter 2 in random Cayley graphs and demonstrates their optimality, with examples comparing their speed to Erdős-Rényi graphs.

Findings

Threshold p for diameter ≤ 2 is approximately √((2+ε) log n / n).

Threshold p for diameter > 2 is approximately √((1/4+ε) log n / n).

Some Cayley graphs reach diameter 2 faster than Erdős-Rényi graphs.

Abstract

Given a group G, the model $\mathcal{G}(G,p)$ denotes the probability space of all Cayley graphs of G where each element of the generating set is chosen independently at random with probability p. In this article we show that for any $\epsilon > 0$ and any family of groups G_k of order n_k for which $n_k \to \infty$, a graph $\Gamma_k \in \mathcal{G}(G_k,p)$ with high probability has diameter at most 2 if $p \geqslant \sqrt{(2 + \epsilon) \frac{\log{n_k}}{n_k}}$ and with high probability has diameter greater than 2 if $p \leqslant \sqrt{(1/4 + \epsilon)\frac{\log{n_k}}{n_k}}$. We also provide examples of families of graphs which show that both of these results are best possible. Of particular interest is that for some families of groups, the corresponding random Cayley graphs achieve diameter 2 significantly faster than the Erd\H{o}s-Renyi random graphs.