On generalized competition index of a primitive tournament

For positive integers k and m and a digraph D , the k -step m -competition graph C m k ( D ) of D has the same set of vertices as D and an edge between vertices x and y if and only if there exist m distinct vertices v 1 , v 2 , … , v m in D such that there exist directed walks of length k from x to v i and from y to v i for 1 ≤ i ≤ m . The m -competition index of a primitive digraph D is the smallest positive integer k such that C m k ( D ) is a complete graph. In this paper, we study the m -competition indices of primitive tournaments and provide an upper bound for the m -competition index of a primitive tournament.