Characterize extremal graphs at lower regularity

Prove that, under an appropriate connectivity or Hamiltonicity assumption, extremal graphs for the random-induced-subgraph Hamiltonicity problem with degree $d=cm$ for fixed $c\in(0,1/2)$ are essentially disjoint unions of bipartite graphs with a bounded number of additional edges enforcing the assumption.

Background

The authors discuss lowering the regularity degree from the Dirac-scale value while imposing an additional condition such as kk-connectivity or Hamiltonicity, since without such a condition disconnected graphs give exponentially small probabilities. They conjecture that the extremal examples should be essentially disjoint unions of bipartite graphs, with only a few added edges needed to ensure connectivity or Hamiltonicity. They also formulate a weaker quantitative version concerning the order of the Hamiltonicity probability.

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