PreprintCC BY 4.0arXiv
Hitting time for loose Hamilton cycles in the random hypergraph process
Abstract
The random hypergraph process starts with an empty hypergraph on n vertices and adds a uniformly chosen missing edge at each step. For even n, we prove that, with high probability, a $3$-uniform loose Hamilton cycle appears when the last isolated vertex disappears. The approach developed here adapts arguments from Kahn's work on perfect matchings. It can also be extended to obtain the corresponding hitting-time theorems for r=2 and every fixed r≥4 satisfying (r−1)| n. Furthermore, it can be used to estimate the numbers of loose Hamilton cycles at these hitting times for every fixed r≥2, up to a factor of exp{o(n)}.
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BibTeX
@misc{leon2026hitting,
title = {{Hitting time for loose Hamilton cycles in the random hypergraph process}},
author = {Jared León and Xizhi Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29894},
eprint = {2610.29894},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29894},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Hitting time for loose Hamilton cycles in the random hypergraph process
AU - Jared León
AU - Xizhi Liu
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29894
UR - https://arxiv.org/abs/2610.29894
AB - The random hypergraph process starts with an empty hypergraph on n vertices and adds a uniformly chosen missing edge at each step. For even n, we prove that, with high probability, a $3$-uniform loose Hamilton cycle appears when the last isolated vertex disappears. The approach developed here adapts arguments from Kahn's work on perfect matchings. It can also be extended to obtain the corresponding hitting-time theorems for r=2 and every fixed r≥4 satisfying (r−1)| n. Furthermore, it can be used to estimate the numbers of loose Hamilton cycles at these hitting times for every fixed r≥2, up to a factor of exp{o(n)}.
ER -