Sharp threshold for the square of a Hamilton cycle

Determine whether the sharp threshold for the existence of the square of a Hamilton cycle in the binomial random graph G(n,p) is e/n^1/2.

Background

The paper discusses the threshold for containing the second power, or square, of a Hamilton cycle in the binomial random graph G(n,p). Earlier work established the correct order of magnitude as Theta(n{-1/2}), but the precise constant was conjectured to be e1/2, yielding the threshold e/n1/2. The paper itself establishes the sharp threshold only for powers k4, so this cited conjecture concerning the square is not resolved by the stated main theorem.

References

They conjectured that the sharp threshold should be \sqrt{e/n} --- a hypothesis that has attracted significant interest in the community.

— Sharp thresholds for higher powers of Hamilton cycles in random graphs  (2502.14515 - Makai et al., 20 Feb 2025) in Section 1, Introduction