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arXiv · 2202.05215

The square of a Hamilton cycle in randomly perturbed graphs

Julia Böttcher·Olaf Parczyk·Amedeo Sgueglia·Jozef Skokan
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Abstract

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.

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Julia Böttcher, Olaf Parczyk, Amedeo Sgueglia, Jozef Skokan. 2022-02-10. The square of a Hamilton cycle in randomly perturbed graphs. https://doi.org/10.1002/rsa.21215

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