Cycle-pancyclism in bipartite tournaments I

Let T be a hamiltonian bipartite tournament with n vertices, γ a hamiltonian directed cycle of T, and k an even number. In this paper, the following question is studied: What is the maximum intersection with γ of a directed cycle of length k? It is proved that for an even k in the range 4 ≤ k ≤ [(n+4)/2], there exists a directed cycle $C_{h(k)}$ of length h(k), h(k) ∈ {k,k-2} with $|A(C_{h(k)}) ∩ A(γ)| ≥ h(k)-3$ and the result is best possible.
In a forthcoming paper the case of directed cycles of length k, k even and k < [(n+4)/2] will be studied.