On the trace of random walks on random graphs

TL;DR

This paper investigates the properties of the trace of a random walk on a random graph, demonstrating that it becomes Hamiltonian and highly connected after a certain number of steps, with specific results for complete graphs.

Contribution

It establishes that the trace of a random walk on a random graph becomes Hamiltonian and well-connected after a precise number of steps, extending understanding of graph properties during random walks.

Findings

Trace becomes Hamiltonian after (1+ε)n log n steps for p > C log n / n

Trace is Θ(log n)-connected at this stage

In complete graphs, the trace becomes Hamiltonian immediately after the last vertex visit

Abstract

We study graph-theoretic properties of the trace of a random walk on a random graph. We show that for any $\varepsilon>0$ there exists $C>1$ such that the trace of the simple random walk of length $(1+\varepsilon)n\ln{n}$ on the random graph $G\sim G(n,p)$ for $p>C\ln{n}/n$ is, with high probability, Hamiltonian and $\Theta(\ln{n})$-connected. In the special case $p=1$ (i.e. when $G=K_n$), we show a hitting time result according to which, with high probability, exactly one step after the last vertex has been visited, the trace becomes Hamiltonian, and one step after the last vertex has been visited for the $k$'th time, the trace becomes $2k$-connected.