t-Pancyclic Arcs in Tournaments

Let T be a non-trivial tournament. An arc is \emph{t-pancyclic} in T, if it is contained in a cycle of length ℓ for every t≤ℓ≤|V(T)|. Let pt(T) denote the number of t-pancyclic arcs in T and ht(T) the maximum number of t-pancyclic arcs contained in the same Hamiltonian cycle of T. Moon ( J. Combin. Inform. System Sci., 19 (1994), 207-214) showed that h3(T)≥3 for any non-trivial strong tournament T and characterized the tournaments with h3(T)=3. In this paper, we generalize Moon s theorem by showing that ht(T)≥t for every 3≤t≤|V(T)| and characterizing all tournaments which satisfy ht(T)=t. We also present all tournaments which fulfill pt(T)=t.