Asymptotic order of hitting numbers for dense regular graphs below the constant-boundary region

Determine the asymptotic order of D_{c,rho}(n), the maximum hitting number among n-vertex regular graphs with independence ratio at least c and degree at least rho n, throughout the region 2c+rho less than or equal to 1; in particular, determine whether D_{c,rho}(n) is bounded by a constant depending only on c and rho whenever c>1/4, and whether it satisfies a bound of order sqrt(n) times a polylogarithmic factor throughout the region.

Background

The function D_{c,rho}(n) is defined as the maximum of h(G) over regular n-vertex graphs satisfying alpha(G) at least cn and degree at least rho n. The paper proves a constant bound when 2c+rho>1 and a logarithmic bound when c>1/4.

The unresolved region 2c+rho<=1 is not uniformly understood. Joined shift-graph constructions and augmented switching graphs show that some admissible parameter pairs have hitting number of order at least sqrt(n), ruling out a universal o(sqrt(n)) estimate. The authors ask whether the logarithmic regime above c=1/4 can be improved to a constant bound and whether a general square-root-with-polylogarithmic-factor upper bound holds.

References

Determine the asymptotic order of $D_{c,\rho}(n)$ in the region $2c+\rho\le1$. In particular, if $c>1/4$, must $D_{c,\rho}(n)$ be bounded by a constant depending only on $c$ and $\rho$? More generally, throughout this region, does $D_{c,\rho}(n)=O_{c,\rho}\bigl(\sqrt n\,(\log n){K(c,\rho)}\bigr)$ hold for some constant $K(c,\rho)$ whenever the eligible family is nonempty?

— Hitting Maximum Independent Sets in Dense and Highly Connected Graphs  (2608.18963 - Bai et al., 19 Aug 2026) in Problem 1, Section 6 (Open Problems)