Bound the square-cycle threshold by a local path-coverage threshold

Determine whether there exists a fixed constant k\in\mathbb{N} such that, for F equal to the square of a cycle on [n], p_c(F)\leq p_e(F)+p_c(\mathcal{Q}_k), where \mathcal{Q}_k is the property that every vertex belongs to the square of a path of length k; in particular, determine whether the inequality holds for k=2, when \mathcal{Q}_2 is the property that the graph has no isolated vertices.

Background

The paper proves the sharp threshold p_c(F)=(1+o(1))\sqrt{e/n} for the square of a Hamilton cycle using a fragmentation and fragment-smoothing method. The authors explain that analogous hitting-time statements fail for denser regular spanning graphs, motivating a weaker comparison with a local property.

The proposed inequality would relate the global threshold for containing a square of a Hamilton cycle to the expectation threshold and the threshold for ensuring that every vertex lies in a bounded-length squared path. The authors explicitly leave open whether any constant k suffices, and single out k=2 as a particularly natural case.

References

On the other hand, our proof method suggests that the following question may have a positive answer: Is there a (small) constant $k\in\mathbb{N}$ such that $p_c(F)\leq p_e(F)+p_c(\mathcal{Q}_k)$, where $F$ is the square of a cycle on $[n]$ and $\mathcal{Q}_k$ denotes the property that every vertex belongs to the square of a path of length $k$? In particular, is it true for $k=2$, i.e. when $\mathcal{Q}_k$ is the property that there are no isolated vertices?

— Sharp thresholds for spanning regular subgraphs  (2502.14794 - Zhukovskii, 20 Feb 2025) in Section 6, “Remaining challenges”