The hitting time of nice factors
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910659634003968 |
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| author | Burghart, Fabian Kaufmann, Marc Müller, Noela Pasch, Matija |
| author_facet | Burghart, Fabian Kaufmann, Marc Müller, Noela Pasch, Matija |
| contents | Consider the random $u$-uniform hypergraph (or $u$-graph) process on $n$ vertices, where $n$ is divisible by $r>u\ge 2$. It was recently shown that with high probability, as soon as every vertex is covered by a copy of the complete $u$-graph $K_r$, it also contains a $K_r$-factor (RSA, Vol. 65 II, Sept. 2024). The hitting time result is obtained using a process coupling, which is based on the proof of the corresponding sharp threshold result (RSA, Vol. 61 IV, Dec. 2022). The latter, however, was not only derived for complete $u$-graphs, but for a broader class of so-called nice $u$-graphs.
The purpose of this article is to extend the process coupling for complete $u$-graphs to the full scope of the sharp threshold result: nice $u$-graphs. As a byproduct, we obtain the extension of the hitting time result to nice $u$-graphs. Since the relevant combinatorial bounds in the proof for the $K_r$-case cannot be generalized, we introduce new arguments that do not only apply to nice u-graphs, but will be relevant for the broader class of strictly 1-balanced u-graphs. Further, we show how the remainder of the process coupling for the $K_r$-case can be utilized in a black-box manner for any u-graph. These advances pave the way for future generalizations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_17764 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The hitting time of nice factors Burghart, Fabian Kaufmann, Marc Müller, Noela Pasch, Matija Combinatorics Probability 05C70 (primary), 05C80, 05C65, 60C05 (secondary) Consider the random $u$-uniform hypergraph (or $u$-graph) process on $n$ vertices, where $n$ is divisible by $r>u\ge 2$. It was recently shown that with high probability, as soon as every vertex is covered by a copy of the complete $u$-graph $K_r$, it also contains a $K_r$-factor (RSA, Vol. 65 II, Sept. 2024). The hitting time result is obtained using a process coupling, which is based on the proof of the corresponding sharp threshold result (RSA, Vol. 61 IV, Dec. 2022). The latter, however, was not only derived for complete $u$-graphs, but for a broader class of so-called nice $u$-graphs. The purpose of this article is to extend the process coupling for complete $u$-graphs to the full scope of the sharp threshold result: nice $u$-graphs. As a byproduct, we obtain the extension of the hitting time result to nice $u$-graphs. Since the relevant combinatorial bounds in the proof for the $K_r$-case cannot be generalized, we introduce new arguments that do not only apply to nice u-graphs, but will be relevant for the broader class of strictly 1-balanced u-graphs. Further, we show how the remainder of the process coupling for the $K_r$-case can be utilized in a black-box manner for any u-graph. These advances pave the way for future generalizations. |
| title | The hitting time of nice factors |
| topic | Combinatorics Probability 05C70 (primary), 05C80, 05C65, 60C05 (secondary) |
| url | https://arxiv.org/abs/2409.17764 |