Determine the extremal graphs for lower-degree robust Hamiltonicity

Determine whether, for fixed \(c\in(0,1/2)\) and sufficiently large \(m\), the extremal examples for the probability that a random induced subgraph of a Hamiltonian or suitably connected \(d\)-regular graph is Hamiltonian are essentially disjoint unions of bipartite graphs with a few edges added to ensure the required connectivity or Hamiltonicity.

Background

The paper proposes varying the number of vertices, regularity degree, and vertex-inclusion probability in the robust Hamiltonicity problem. For degrees below half the number of vertices, disconnected graphs create exponentially small probabilities, so the authors suggest imposing connectivity or Hamiltonicity assumptions. They conjecture that the extremal structures under such assumptions should resemble disjoint unions of bipartite graphs with a small number of additional edges.