Optimal budget for powers of Hamilton cycles

Close the gap between the lower and upper bounds on the budget required for successful strategies constructing powers of Hamilton cycles in the budget-constrained random graph process.

Background

The paper proves nearly matching lower and upper bounds for powers of Hamilton cycles, but the bounds differ by subpolynomial factors. The authors explicitly identify closing this gap as an unresolved direction.

References

In a similar direction to \cref{prob:close_gap}, it would be interesting to close the gap between the lower and the upper bound for powers of Hamilton cycles (\cref{thm:hamsquare_lower,thm:hamsquare_upper}).

— Graph factors and powers of Hamilton cycles in the budget-constrained random graph process  (2502.20870 - Díaz et al., 28 Feb 2025) in Section 7 “Final remarks and open problems”