Monochromatic paths and monochromatic sets of arcs in 3-quasitransitive digraphs

We call the digraph D an m-coloured digraph if the arcs of D are coloured with m colours. 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 for every pair of vertices of N there is no monochromatic path between them and for every vertex v / ∈ N there is a monochromatic path from v to N . We denote by A(u) the set of arcs of D that have u as the initial vertex. We prove that if D is an m-coloured 3-quasitransitive digraph such that for every vertex u of D, A(u) is monochromatic and D satisfies some colouring conditions over one subdigraph of D of order 3 and two subdigraphs of D of order 4, then D has a kernel by monochromatic paths.