Acquaintance time of random graphs near connectivity threshold

TL;DR

This paper proves a conjecture about the acquaintance time of random graphs near the connectivity threshold, showing it is asymptotically bounded by a function of the edge probability at the moment the graph becomes connected, and extends the results to hypergraphs.

Contribution

It confirms the conjecture on acquaintance time at the connectivity threshold and generalizes the results to random hypergraphs.

Findings

Acquaintance time is bounded by p^{-1} log^{O(1)} n at connectivity.

The conjecture holds exactly when the graph becomes connected.

Results are extended to hypergraph models.

Abstract

Benjamini, Shinkar, and Tsur stated the following conjecture on the acquaintance time: asymptotically almost surely $AC(G) \le p^{-1} \log^{O(1)} n$ for a random graph $G \in G(n,p)$, provided that $G$ is connected. Recently, Kinnersley, Mitsche, and the second author made a major step towards this conjecture by showing that asymptotically almost surely $AC(G) = O(\log n / p)$, provided that $G$ has a Hamiltonian cycle. In this paper, we finish the task by showing that the conjecture holds in the strongest possible sense, that is, it holds right at the time the random graph process creates a connected graph. Moreover, we generalize and investigate the problem for random hypergraphs.