On generalized competition index of a primitive tournament

AbstractFor positive integers k and m and a digraph D, the k-step m-competition graph Cmk(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 v1,v2,…,vm in D such that there exist directed walks of length k from x to vi and from y to vi for 1≤i≤m. The m-competition index of a primitive digraph D is the smallest positive integer k such that Cmk(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