Polynomial lower bound for lower-degree regular graphs

Prove that if d=cm for a fixed c in (0,1/2) and m is sufficiently large, then the Hamiltonicity probability p(G) for a random induced subgraph of an appropriate d-regular graph G satisfies $p(G)=\Omega(m^{-k/2})$, where $k=\lfloor(2c)^{-1}\rfloor$, under the additional connectivity or Hamiltonicity assumption needed for the generalized problem.

Background

As a more explicit and weaker version of their structural conjecture, the authors formulate a quantitative lower-bound problem for d-regular graphs with d=cm and c<1/2. The exponent depends on the number of vertices that must be balanced across the approximately bipartite structure, represented by k=⌊(2c)−1⌋k=\lfloor(2c)^{-1}\rfloor.

This problem seeks a polynomial, rather than exponential, lower bound on the probability that a random induced subgraph is Hamiltonian. It is presented as a concrete target arising from the broader conjecture about extremal graph structure.

References

We conjecture that the extremal examples $G$ for such questions are essentially disjoint unions of bipartite graphs, with a few edges added to ensure the connectivity or Hamiltonicity assumption. A more explicit and weaker form of this conjecture, which still seems interesting, would be to show that if $d=cm$ for fixed $c \in (0,1/2)$ and $m$ large then $p(G) = \Omega(m{-k/2})$ where $k = \lfloor (2c){-1} \rfloor$.

— Cyclic subsets in regular Dirac graphs  (2503.01826 - Draganić et al., 3 Mar 2025) in Section 6, Concluding remarks