The square of a Hamilton cycle in randomly perturbed graphs
| Autor: | Böttcher, Julia, Parczyk, Olaf, Sgueglia, Amedeo, Skokan, Jozef |
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| Rok vydání: | 2022 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | We investigate the appearance of the square of a Hamilton cycle in the model of randomly perturbed graphs, which is, for a given $\alpha \in (0,1)$, the union of any $n$-vertex graph with minimum degree $\alpha n$ and the binomial random graph $G(n,p)$. This is known when $\alpha > 1/2$, and we determine the exact perturbed threshold probability in all the remaining cases, i.e., for each $\alpha \le 1/2$. We demonstrate that, as $\alpha$ ranges over the interval $(0,1)$, the threshold performs a countably infinite number of `jumps'. Our result has implications on the perturbed threshold for $2$-universality, where we also fully address all open cases. Comment: 39 pages, 8 figures; final version as accepted for publication in Random Structures & Algorithms |
| Databáze: | arXiv |
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