The number of pancyclic arcs in a k-strong tournament

A tournament is a digraph, where there is precisely one arc between every pair of distinct vertices. An arc is pancyclic in a digraph D, if it belongs to a cycle of length l, for all 3 <= l <= |V (D)|. Let p(D) denote the number of pancyclic arcs in a digraph D and let h(D) denote the maximum number of pancyclic arcs belonging to the same Hamilton cycle of D. Note that p(D) >= h(D). Moon showed that h(T ) >= 3 for all strong nontrivial tournaments, T , and Havet showed that h(T ) >= 5 for all 2-strong tournaments T . We will in this talk show that if T is a k-strong tournament, with k >= 2, then p(T ) >= nk/2 and h(T ) >= (k + 5)/2. This solves a conjecture by Havet, stating that there exists a constant ak, such that p(T ) >= ak ∗ n, for all k-strong tournaments, T , with k >= 2. Furthermore the second result gives support for the conjecture h(T ) >= 2k + 1, which was also stated by Havet. The previously best known bounds when k >= 2 were p(T ) >= 2k + 3 and h(T ) >= 5. Furthermore some of the lemma’s used in the above proofs immediately imply that every regular tournament is arc-pancyclic (which was first proved by Alspach), and that every 2-strong tournament contains 2 distinct vertices, such that all arcs out of them are arc-pancyclic. We conjecture that there in fact exists 3 such vertices, which would be best possible (even if we looked at k-strong tournaments for any fixed k > 1).