Completeness of strongly negative obstructions to Hamiltonicity in G(n,W)

Determine whether the following four conditions are a complete characterization of graphons W for which the inhomogeneous random graph G(n,W) is asymptotically almost surely non-Hamiltonian: (1) W is not connected; (2) the small-degree tail diverges, i.e., lim_{alpha→0} μ({x ∈ Ω : deg_W(x) ≤ α})/α = ∞; (3) W has a narrow peninsula, meaning there exist disjoint measurable sets A,B ⊂ Ω and a ∈ (0,1/2] with μ(A) > a, μ(B) = 1−2a, and W = 0 almost everywhere on A × (A ∪ B); and (4) there exists a partition Ω = S ⊔ T with μ(S) = μ(T) = 1/2 such that W = 0 almost everywhere on (S × S) ∪ (T × T). Specifically, show that if W violates none of these four conditions, then limsup_{n→∞} P[G(n,W) is Hamiltonian] > 0.

Background

The paper proves an if-and-only-if criterion for asymptotically almost sure Hamiltonicity of G(n,W) based on three properties of the graphon W: connectedness, a light small-degree tail, and the absence of a peninsula. In the concluding remarks, the authors turn to the opposite regime: identifying graphons for which G(n,W) is asymptotically almost never Hamiltonian.

They present four explicit “strongly negative” conditions which each imply that G(n,W) is a.a.s. non-Hamiltonian: (i) W is disconnected; (ii) the proportion of points of degree at most α decays too slowly (the ratio μ(D_W(α))/α diverges); (iii) W contains a narrow peninsula, a structural obstruction that prevents perfect fractional matchings; and (iv) W admits a balanced bipartition with no within-part edges, forcing a near-balanced bipartite structure that obstructs Hamiltonicity.

The open problem asks whether these four conditions are complete for the “a.a.s. never Hamiltonian” regime: namely, if none of them holds, does the Hamiltonicity probability stay bounded away from zero along some subsequence (limsup > 0)? This would sharply separate graphons into those that almost surely never yield a Hamiltonian graph and those for which Hamiltonicity occurs with positive probability in the limit.