Cycle-pancyclism in tournaments I

LetT be a hamiltonian tournament withn vertices andγ a hamiltonian cycle ofT. In this paper we start the study of the following question: What is the maximum intersection withγ of a cycle of lengthk? This number is denotedf(n, k). We prove that fork in range, 3 ≤k ≤n + 4/2,f(n,k) ≥ k ? 3, and that the result is best possible; in fact, a characterization of the values ofn, k, for whichf(n, k) = k ? 3 is presented. In a forthcoming paper we studyf(n, k) for the case of cycles of lengthk > n + 4/2.