Vertex 6-pancyclic in-tournaments Original Research Article

An in-tournament is a loopless digraph without multiple arcs and cycles of length 2 such that the negative neighborhood of every vertex induces a tournament. This paper tackles the problem of vertex k-pancyclicity in strong in-tournaments of order n, i.e., every vertex belongs to a cycle of length l for every k⩽l⩽n. In 2001, Tewes and Volkmann (J. Graph Theory 36 (2001) 84) gave sharp lower bounds for the minimum degree such that a strong in-tournament is vertex k-pancyclic for k⩽5 and k⩾n−3. In accordance with a family of examples (see J. Graph Theory 36 (2001) 84) and their results, they conjectured that every strong in-tournament of order n with minimum degree greater than 9(n−k−1)5+6k+(−1)k2−k+2+1 is vertex k-pancylic. The main result of this paper is the confirmation of this conjecture for the case k=6 (except for the values n=14,15,16).