Generalized competition index of primitive digraphs

Abstract

For any positive integers k and m, the k-step m-competition graph Cmk(D) of a digraph D has the same set of vertices as D and there is an edge between vertices x and y if and only if there are distinct m vertices v1, v2, · · ·, vm in D such that there are directed walks of length k from x to vi and from y to vi for all 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 obtained some sharp upper bounds for the m-competition indices of various classes of primitive digraphs.