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 ${\displaystyle C_{h\left(k\right)}}$ of length h(k), h(k) ∈ {k,k-2} with ${\displaystyle |A\left(C_{h\left(k\right)}\right)\cap A(\gamma )|\ge h\left(k\right)-3}$ 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.