Clique factors in randomly perturbed graphs: the transition points
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arXiv
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| Format: | Preprint |
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2024
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| author | Antoniuk, Sylwia Kamčev, Nina Reiher, Christian |
| author_facet | Antoniuk, Sylwia Kamčev, Nina Reiher, Christian |
| contents | A randomly perturbed graph $G^p = G_α\cup G(n,p)$ is obtained by taking a deterministic $n$-vertex graph $G_α= (V, E)$ with minimum degree $δ(G)\geq αn$ and adding the edges of the binomial random graph $G(n,p)$ defined on the same vertex set $V$. For which value $p$ (depending on $α$) does the graph $G^p$ contain a $K_r$-factor (a spanning collection of vertex-disjoint $K_r$-copies) with high probability? The order of magnitude of the minimal value of $p$ has been determined whenever $α\neq 1- \frac{s}{r}$ for an integer $s$ (see Han, Morris, and Treglown [RSA, 2021] and Balogh, Treglown, and Wagner [CPC, 2019]).
We establish the minimal probability $p_s$ (up to a constant factor) for all values of $α= 1-\frac{s}{r} \leq \frac 12$, and show that the threshold exhibits a polynomial jump at $α= 1-\frac{s}{r}$ compared to the surrounding intervals. An extremal example $G_α$ which shows that $p_s$ is optimal up to a constant factor differs from the previous (usually multipartite) examples in containing a pseudorandom induced subgraph. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_11003 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Clique factors in randomly perturbed graphs: the transition points Antoniuk, Sylwia Kamčev, Nina Reiher, Christian Combinatorics 05C80, 05D10, 05C55 A randomly perturbed graph $G^p = G_α\cup G(n,p)$ is obtained by taking a deterministic $n$-vertex graph $G_α= (V, E)$ with minimum degree $δ(G)\geq αn$ and adding the edges of the binomial random graph $G(n,p)$ defined on the same vertex set $V$. For which value $p$ (depending on $α$) does the graph $G^p$ contain a $K_r$-factor (a spanning collection of vertex-disjoint $K_r$-copies) with high probability? The order of magnitude of the minimal value of $p$ has been determined whenever $α\neq 1- \frac{s}{r}$ for an integer $s$ (see Han, Morris, and Treglown [RSA, 2021] and Balogh, Treglown, and Wagner [CPC, 2019]). We establish the minimal probability $p_s$ (up to a constant factor) for all values of $α= 1-\frac{s}{r} \leq \frac 12$, and show that the threshold exhibits a polynomial jump at $α= 1-\frac{s}{r}$ compared to the surrounding intervals. An extremal example $G_α$ which shows that $p_s$ is optimal up to a constant factor differs from the previous (usually multipartite) examples in containing a pseudorandom induced subgraph. |
| title | Clique factors in randomly perturbed graphs: the transition points |
| topic | Combinatorics 05C80, 05D10, 05C55 |
| url | https://arxiv.org/abs/2410.11003 |