A conjecture on cycle-pancyclism in tournaments

Let T be a hamiltonian tournament with n vertices and γ a hamiltonian cycle of T . In previous works we introduced and studied the concept of cycle-pancyclism to capture the following question: What is the maximum intersection with γ of a cycle of length k? More precisely, for a cycle Ck of length k in T we denote Iγ(Ck) = |A(γ) ∩ A(Ck)|, the number of arcs that γ and Ck have in common. Let f(k, T, γ) = max{Iγ(Ck)|Ck ⊂ T} and f(n, k) = min{f(k, T, γ)|T is a hamiltonian tournament with n vertices, and γ a hamiltonian cycle of T}. In previous papers we gave a characterization of f(n, k). In particular, the characterization implies that f(n, k) ≥ k − 4. The purpose of this paper is to give some support to the following original conjecture: for any vertex v there exists a cycle of length k containing v with f(n, k) arcs in common with γ.