The generalized competition indices of primitive minimally strong digraphs

Let D be a primitive digraph of order n and 1≤m≤n. The m-competition index of D is the smallest positive integer k such that every pair of vertices x and y of D have at least m common preys in Dk. In this paper, the upper bound of m-competition indices (1≤m≤n−1) for primitive minimally strong digraphs of order n is obtained. Furthermore, it is shown that for 1≤m≤n−1, there exist “gaps” in the m-competition index set of primitive minimally strong digraphs of order n.