A note on the acquaintance time of random graphs

TL;DR

This paper proves the asymptotic behavior of the acquaintance time in random graphs, confirming a conjecture and establishing bounds that relate the graph's density to the acquaintance process.

Contribution

It confirms a conjecture on the acquaintance time of random graphs and provides matching bounds, improving understanding of graph covering and acquaintance processes.

Findings

Asymptotically almost surely, $AC(G) = O(rac{ ext{log} n}{p})$ for $G ext{ in } G(n,p)$ when $pn > (1+ ext{epsilon}) ext{log} n$.

A matching lower bound for dense graphs shows $K_n$ cannot be covered with fewer than $O(rac{ ext{log} n}{p})$ copies of $G$ under certain conditions.

An improved general upper bound states $AC(G) = O(rac{n^2}{ ext{log} n})$ for any $n$-vertex graph.

Abstract

In this short note, we prove the conjecture of Benjamini, Shinkar, and Tsur on the acquaintance time $AC(G)$ of a random graph $G \in G(n,p)$. It is shown that asymptotically almost surely $AC(G) = O(\log n / p)$ for $G \in G(n,p)$, provided that $pn > (1+\epsilon) \log n$ for some $\epsilon > 0$ (slightly above the threshold for connectivity). Moreover, we show a matching lower bound for dense random graphs, which also implies that asymptotically almost surely $K_n$ cannot be covered with $o(\log n / p)$ copies of a random graph $G \in G(n,p)$, provided that $pn > n^{1/2+\epsilon}$ and $p < 1-\epsilon$ for some $\epsilon>0$. We conclude the paper with a small improvement on the general upper bound showing that for any $n$-vertex graph $G$, we have $AC(G) = O(n^2/\log n)$.