γ-cycles in arc-colored digraphs

We call a digraph D an m-colored digraph if the arcs of D are colored with m colors. A directed path (or a directed cycle) is called monochromatic if all of its arcs are colored alike. A subdigraph H in D is called rainbow if all of its arcs have different colors. A set N ⊆ V (D) is said to be a kernel by monochromatic paths of D if it satisfies the two following conditions:for every pair of different vertices u, v ∈ N there is no monochromatic path in D between them, andfor every vertex x ∈ V (D) − N there is a vertex y ∈ N such that there is an xy-monochromatic path in D