On the existence and number of (k+1)-kings in k-quasi-transitive digraphs

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, then (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 3king 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).