Vertex pancyclic in-tournaments

An in-tournament is an oriented graph such that the negative neighborhood o f every vertex induces a tournament. The topic of this paper is to investig ate vertex k-pancyclicity of in-tournaments of order n; where for some 3 le ss than or equal to k less than or equal to n, every vertex belongs to a cy cle of length p for every k less than or equal to p less than or equal to n . We give sharp lower bounds for the minimum degree such that a strong in-t ournament is Vertex k-pancyclic for k less than or equal to 5 and k greater than or equal to n-3. In the latter case, we even show that the in-tournam ents in consideration are fully (n-3)-extendable which means that every ver tex belongs to a cycle of length n-3 and that the vertex set of every cycle of length at least n-3 is contained in a cycle of length one greater. in a ccordance with these results, we state the conjecture that every strong in- tournament of order n with minimum degree greater than 9(n-k-1) / 5+6k+(-1) (k)2(-k divided by2) + 1 is vertex k-pancyclic for 5 < k < n-3, and we pres ent a family of examples showing that this bound would be best possible. (C ) 2001 John Wiley & Sons, Inc.