Note on the trace of random walks on pseudorandom graphs

TL;DR

This paper investigates the properties of the trace of random walks on pseudorandom graphs, establishing bounds on cover time and Hamiltonicity that are optimal and answer open questions in the field.

Contribution

It provides new bounds on cover time and Hamiltonicity of random walk traces on pseudorandom and random regular graphs, resolving previously open questions.

Findings

Cover time is at most (1+ε)n log n with high probability.

The trace of a random walk of this length is Hamiltonian with high probability.

Results hold for large degree random regular graphs.

Abstract

We study the graph-theoretic properties of the trace of random walks on pseudorandom graphs. We show that for any $\varepsilon>0$, there exists a constant $C$ such that the cover time of an $(n,d,\lambda)$-graph $G$ with $d/\lambda\ge C$ is at most $(1+\varepsilon)n\log n$, meaning the expected number of steps needed to reach all vertices at least once is at most $(1+\varepsilon)n\log n$ regardless of the starting vertex. Furthermore, we prove that with high probability, the trace of a random walk of length $(1+\varepsilon)n\log n$ on $G$ is Hamiltonian, regardless of the starting vertex. These results also hold for random $d$-regular graphs with sufficiently large $d$. These findings answer two questions proposed by Frieze, Krivelevich, Michaeli, and Peled [PLMS, 2018]. Notably, our results imply a bound on a stronger version of the cover time: with high probability, all vertices are covered after $(1+\varepsilon)n\log n$ steps, regardless of the starting vertex. Our proofs rely on the spectral properties of the adjacency matrix and the graph expansion. All results are asymptotically optimal.