Determine extremal constructions for lower-degree random induced-subgraph Hamiltonicity

Characterize the extremal examples for the probability \(h(G,1/2)\) that a random induced subgraph of a Hamiltonian or suitably connected \(d\)-regular graph on \(m\) vertices is Hamiltonian when \(d=cm\) with fixed \(c\in(0,1/2)\), and in particular prove the explicit lower bound \(p(G)=\Omega(m^{-k/2})\) with \(k=\lfloor(2c)^{-1}\rfloor\) if that is the intended weaker formulation.

Background

The paper proposes varying both the regularity degree and the vertex-inclusion probability in the random induced-subgraph Hamiltonicity problem. For degrees below the Dirac threshold, an additional assumption such as connectivity, Hamiltonicity, or an analogous structural condition is needed to avoid trivial exponentially small probabilities caused by disconnected graphs.

The authors conjecture that the relevant extremal constructions should essentially be disjoint unions of bipartite graphs, with a small number of additional edges enforcing the required connectivity or Hamiltonicity. They also state a concrete weaker form predicting a polynomial lower bound for fixed density c<1/2c<1/2.

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”