On generalized competition index of a primitive tournament

For positive integers kk and mm and a digraph DD, the kk-step mm-competition graph Cmk(D) of DD has the same set of vertices as DD and an edge between vertices xx and yy if and only if there exist mm distinct vertices v1,v2,…,vmv1,v2,…,vm in DD such that there exist directed walks of length kk from xx to vivi and from yy to vivi for 1≤i≤m1≤i≤m. The mm-competition index of a primitive digraph DD is the smallest positive integer kk such that Cmk(D) is a complete graph. In this paper, we study the mm-competition indices of primitive tournaments and provide an upper bound for the mm-competition index of a primitive tournament.