Hamiltonicity of a regular graph perturbed by a random 2-factor

Prove that for every n-vertex regular graph G and an independently chosen random 2-regular graph F on the same vertex set, the union G ∪ F is Hamiltonian with high probability, potentially without any additional assumptions on the degree or structure of G.

Background

The paper establishes that if an arbitrary graph G has minimum degree at least approximately √(n log n)/2, then adding a random 2-factor produces a Hamiltonian graph with high probability. It also proves a substantially stronger result for approximately regular graphs: minimum degree ω(log³ n) suffices when the maximum degree is O(δ(G)).

The authors identify the remaining issue of removing the approximate-regularity requirement. They conjecture that every regular graph, regardless of its degree and without any further structural assumption, becomes Hamiltonian after the addition of a random 2-factor. They note supporting evidence from perfect matchings, even-regular graphs, and the known Hamiltonicity of the union of two random 2-factors.

References

Considering hamiltonicity of sparse graphs, our final open problem is to improve Theorem 1.3 under the additional assumption that G is regular (not just approximately regular). It may be that no further assumption is required. Conjecture 6.2. Let G be an n-vertex regular graph and F ∼ Gn,2 be a random 2-regular graph on the same vertex set as G. Then whp G ∪ F is Hamiltonian.

— Pósa rotation through a random permutation  (2502.00489 - Draganić et al., 1 Feb 2025) in Concluding remarks, Section 6, Conjecture 6.2, p. 7