Note on the trace of random walks on pseudorandom graphs
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866910019315826688 |
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| author | Chen, Yaobin Wang, Yiting |
| author_facet | Chen, Yaobin Wang, Yiting |
| contents | 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 $(n,d,λ)$-graph $G$ with $d/λ\ge C$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_10970 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Note on the trace of random walks on pseudorandom graphs Chen, Yaobin Wang, Yiting Combinatorics Probability 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 $(n,d,λ)$-graph $G$ with $d/λ\ge C$ 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. |
| title | Note on the trace of random walks on pseudorandom graphs |
| topic | Combinatorics Probability |
| url | https://arxiv.org/abs/2602.10970 |