Tight Staircase Bounds for Cyclic Subsets below Dirac's Threshold

Let $\operatorname{Cyc}(G)$ denote the number of cyclic subsets in a graph $G$, which are subsets that induce a Hamiltonian subgraph. Dragani\'{c}, Keevash and M\"{u}yesser recently proved that every regular Dirac graph has $\Omega(2^n)$ cyclic subsets, resolving a problem of Erd\H{o}s and Faudree. We determine the sharp asymptotic lower bound throughout the linear range below Dirac's threshold. Let $G$ be an $n$-vertex $d$-regular graph with $d=\Omega(n)$ and $d<n/2$, then $$\operatorname{Cyc}(G)\ge (q-o(1))2^{n/q}, \quad \text{where } \quad q=\left\lfloor \frac{n}{d+1}\right\rfloor \ge 2.$$ This bound is asymptotically best possible, including the leading coefficient $q$, as witnessed at the staircase levels by the disjoint union of $q$ equal cliques. Consequently, the optimal exponential rate changes by discrete jumps as $d$ crosses the thresholds $n/k$, rather than varying smoothly with $d$. We also prove the optimal exponential rate at the Dirac boundary: every $n$-vertex $n/2$-regular graph satisfies $\operatorname{Cyc}(G)\ge 2^{(1-o(1))n},$ which is sharp up to a subexponential factor by $K_{n/2,n/2}$.