Guaranteed overlap between a 2-factor and a Hamilton cycle

Let G be a Hamiltonian graph on n vertices and let F be a 2-factor of G. How many edges of F can one guarantee that some Hamilton cycle of G shares with it? Write ov(n) for the largest integer s such that every pair (G,F) with |V(G)|=n satisfies max_H |F ∩ H| ≥ s, where H ranges over the Hamilton cycles of G. We prove the following. First, a reduction theorem: the extremal problem is concentrated on the graphs that are the union of a 2-factor and a Hamilton cycle, hence have at most 2n edges and maximum degree 4. This makes exact computation feasible, and we obtain ov(n)=4 for 6 ≤ n ≤ 8 and ov(n)=5 for 9 ≤ n ≤ 13, in the last case by traversing all 438,263,364 labelled 2-regular graphs on 13 vertices. Second, we construct two explicit infinite families, Γ_n and Δ_n, defined for every n ≥ 10 by a zigzag over an interval, prove that both are uniquely Hamiltonian, and deduce ov(n) ≤ 5 for every n ≥ 10. For n=10 and n=11 these are, up to the dihedral symmetry, the only uniquely Hamiltonian extremal pairs, while for n=12 there is one more and for n=13 two more. All of them, at every size, have two vertices of degree 2 and six of degree 3, so that these graphs are uniquely Hamiltonian with maximum degree 4 and only eight vertices of smaller degree, uniformly in n. All of them also share a finer invariant: F ∩ H splits into exactly three paths, of one, two and two edges, which by a degree count is what fixes the profile. Third, we show that the reverse inequality is not elementary: even the assertion ov(n) ≥ 1 implies Sheehan's 1975 conjecture on second Hamilton cycles in 4-regular graphs. We also prove a protected-edge lemma: if H attains the maximum, then any competing Hamilton cycle can only give up edges of F ∩ H, so that in an optimal pair the whole Hamiltonian structure is confined to those |F ∩ H| edges and the number of Hamilton cycles is bounded by a function of |F ∩ H| alone, independently of n. Finally, we reduce ov(n) ≥ 5 to the configurations sharing at most four edges with the cycle and, by means of an amplification lemma and a contraction producing a 4-regular multigraph with forbidden transitions, we place the conjecture ov(n)=5 between Sheehan's conjecture and a strengthening of it localised at four vertices.