Packing and finding paths in sparse random graphs

Abstract

Let $G\sim G(n,p)$ be a (hidden) Erd\H{o}s-R\'enyi random graph with $p=(1+ \varepsilon)/n$ for some fixed constant $ \varepsilon >0$. Ferber, Krivelevich, Sudakov, and Vieira showed that to reveal a path of length $\ell=\Omega\left(\frac{\log(1/ \varepsilon)}{ \varepsilon}\right)$ in $G$ with high probability, one must query the adjacency of $\Omega\left(\frac{\ell}{p \varepsilon\log(1/ \varepsilon)}\right)$ pairs of vertices in $G$, where each query may depend on the outcome of all previous queries. Their result is tight up to the factor of $\log(1/ \varepsilon)$ in both $\ell$ and the number of queries, and they conjectured that this factor could be removed. We confirm their conjecture. The main ingredient in our proof is a result about path-packings in random labelled trees of independent interest. Using this, we also give a partial answer to a related question of Ferber, Krivelevich, Sudakov, and Vieira. Namely, we show that when $\ell=o\left((t/\log t)^{1/3}\right)$, the maximum number of vertices covered by edge-disjoint paths of length at least $\ell$ in a random labelled tree of size $t$ is $\Theta(t/\ell)$ with high probability.