PreprintCC BY-NC-ND 4.0arXiv
Sprinkling for random regular graphs and Kim-Vu's sandwich conjecture
Abstract
We show that the uniform random d-regular graph model on n vertices is, in total variation distance, asymptotically close to the union of d uniformly random edge-disjoint perfect matchings, provided that n is even and ω(1) = d ≤ n1/10. This confirms a conjecture by Hollom, Lichev, Mond, Portier, and Wang for this range of parameters. This result has a couple of interesting consequences. First, this shows that random d-regular graphs on n vertices exhibit sprinkling properties for some range of parameters, thereby confirming a conjecture by Isaev, McKay, Southwell and Zhukovskii in that range. Second, we give a short alternative proof of the upper side of Kim-Vu's sandwich conjecture in the sparse regime.
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BibTeX
@misc{portier2026sprinkling,
title = {{Sprinkling for random regular graphs and Kim-Vu's sandwich conjecture}},
author = {Julien Portier},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28395},
eprint = {2610.28395},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28395},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Sprinkling for random regular graphs and Kim-Vu's sandwich conjecture
AU - Julien Portier
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28395
UR - https://arxiv.org/abs/2610.28395
AB - We show that the uniform random d-regular graph model on n vertices is, in total variation distance, asymptotically close to the union of d uniformly random edge-disjoint perfect matchings, provided that n is even and ω(1) = d ≤ n^1/10. This confirms a conjecture by Hollom, Lichev, Mond, Portier, and Wang for this range of parameters. This result has a couple of interesting consequences. First, this shows that random d-regular graphs on n vertices exhibit sprinkling properties for some range of parameters, thereby confirming a conjecture by Isaev, McKay, Southwell and Zhukovskii in that range. Second, we give a short alternative proof of the upper side of Kim-Vu's sandwich conjecture in the sparse regime.
ER -