Establish the lower-degree induced-Hamiltonicity bound

Prove that if \(d=cm\) for fixed \(c\in(0,1/2)\) and \(m\) is sufficiently large, then the probability \(p(G)\) that a random induced subgraph of the relevant \(d\)-regular graph is Hamiltonian satisfies \(p(G)=\Omega(m^{-k/2})\), where \(k=\lfloor(2c)^{-1}\rfloor\).

Background

After discussing lower-degree analogues of the main problem, the paper gives a more explicit and weaker target related to its conjectured extremal structures. The proposed estimate predicts polynomial decay in the number of vertices, with the exponent determined by the degree density. This statement is presented as an unresolved direction rather than proved in the paper.