Every n-vertex d-regular graph with d < n/2 has at least (q-o(1))2^{n/q} cyclic subsets, where q = floor(n/(d+1)), and this bound is asymptotically tight.
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Tight Staircase Bounds for Cyclic Subsets below Dirac's Threshold
Every n-vertex d-regular graph with d < n/2 has at least (q-o(1))2^{n/q} cyclic subsets, where q = floor(n/(d+1)), and this bound is asymptotically tight.