Extremal structure for lower-degree Hamiltonian induced-subgraph problems

Prove that, for fixed c in (0,1/2) and sufficiently large m, the extremal graphs for Hamiltonicity probabilities of random induced subgraphs of degree d=cm are essentially disjoint unions of bipartite graphs, subject to the additional connectivity or Hamiltonicity conditions required to make the problem nondegenerate.

Background

The paper explains that lowering the regularity degree while retaining p=1/2 requires an additional assumption, such as k-connectivity or Hamiltonicity, because otherwise disconnected graphs yield exponentially small Hamiltonicity probabilities. It then proposes a structural conjecture for the extremal examples in this generalized setting.

The conjecture concerns the robust analogue of the original problem for d=cm with c<1/2. The authors expect extremal graphs to be close to disjoint unions of bipartite graphs, with only a few additional edges needed to enforce connectivity or Hamiltonicity.