The square of a Hamilton cycle in randomly perturbed graphs

TL;DR

This paper determines the exact threshold probability for the appearance of the square of a Hamilton cycle in randomly perturbed graphs across all minimum degree ratios, revealing a complex threshold behavior with infinitely many jumps.

Contribution

It provides the complete characterization of the perturbed threshold for the square of a Hamilton cycle for all minimum degree ratios, including the case when   1/2, and addresses open cases in 2-universality.

Findings

Exact threshold probabilities are identified for all   1/2 cases.

The threshold exhibits countably infinite jumps as  varies.

Implications are shown for the 2-universality property in perturbed graphs.

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.