An s-strong tournament with s≥3 has s + 1 vertices whose out-arcs are 4-pancyclic
Discrete Applied Mathematics
Note: The number of vertices whose out-arcs are pancyclic in a 2-strong tournament
Discrete Applied Mathematics
Out-arc pancyclicity of vertices in tournaments
Discrete Applied Mathematics
The structure of 4-strong tournaments containing exactly three out-arc pancyclic vertices
Journal of Graph Theory
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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 non-trivial tournaments, T, and Havet showed that h(T) ≥ 5 for all 2-strong tournaments T. We will show that if T is a k-strong tournament, with k ≥ 2, then p(T) ≥ 1-2, nk and h(T) ≥ (k + 5)-2. This solves a conjecture by Havet, stating that there exists a constant αk, such that p(T) ≥ αk n, for all k-strong tournaments, T, with k ≥ 2. Furthermore, the second results 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. © 2005 Wiley Periodicals, Inc. J Graph Theory