Monochromatic paths and monochromatic sets of arcs in quasi-transitive digraphs

Let D be a digraph, V (D) and A(D) will denote the sets of vertices and arcs of D, respectively. We call the digraph D an m-coloured digraph if each arc of D is coloured by an element of {1, 2, . . . , m} where m ≥ 1. A directed path is called monochromatic if all of its arcs are coloured alike. A set N of vertices of D is called a kernel by monochromatic paths if there is no monochromatic path between two vertices of N and if for every vertex v not in N there is a monochromatic path from v to some vertex in N . A digraph D is called a quasi-transitive digraph if (u, v) ∈ A(D) and (v, w) ∈ A(D) implies (u, w) ∈ A(D) or (w, u) ∈ A(D). We prove that if D is an m-coloured quasi-transitive digraph such that for every vertex u of D the set of arcs that have u as initial end point is monochromatic and D contains no C3 (the 3-coloured directed cycle of length 3), then D has a kernel by monochromatic paths.