On the existence and number of [formula omitted]-kings in [formula omitted]-quasi-transitive digraphs

Abstract

Let D=(V(D),A(D)) be a digraph and k≥2 an integer. We say that D is k-quasi-transitive if for every directed path (v0,v1,…,vk) in D we have (v0,vk)∈A(D) or (vk,v0)∈A(D). Clearly, a 2-quasi-transitive digraph is a quasi-transitive digraph in the usual sense.Bang-Jensen and Gutin proved that a quasi-transitive digraph D has a 3-king if and only if D has a unique initial strong component, and if D has a 3-king and the unique initial strong component of D has at least three vertices, then D has at least three 3-kings. In this paper we prove the following generalization: a k-quasi-transitive digraph D has a (k+1)-king if and only if D has a unique initial strong component, and if D has a (k+1)-king, then either all the vertices of the unique initial strong components are (k+1)-kings or the number of (k+1)-kings in D is at least (k+2). Also, we obtain new results on the minimum number of 3-kings in quasi-transitive digraphs.