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 ${\displaystyle I_{\gamma }(Cₖ)=|A(\gamma )\cap A(Cₖ)|}$, the number of arcs that γ and Cₖ have in common. Let ${\displaystyle f(k,T,\gamma )=max\{I_{\gamma }(Cₖ)\mid Cₖ\subset T\}}$ 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.