Kernels in monochromatic path digraphs

We call the digraphD anm-coloured digraph if its arcs are coloured withm colours. A directed path (or a directed cycle) is called monochromatic if all of its arcs are coloured alike. Let D be an m-coloured digraph. A set N ⊆ V (D) is said to be a kernel by monochromatic paths if it satisfies the following two conditions: (i) for every pair of different vertices u, v ∈ N there is no monochromatic directed path between them and (ii) for each vertex x ∈ (V (D)−N) there is a vertex y ∈ N such that there is an xy-monochromatic directed path. In this paper is defined the monochromatic path digraph ofD, MP (D), and the inner m-colouration of MP (D). Also it is proved that if D is an m-coloured digraph without monochromatic directed cycles, then the number of kernels by monochromatic paths in D is equal to the 408 H. Galeana-Sánchez, L.P. Raḿırez and H.A.R. Mej́ıa number of kernels by monochromatic paths in the inner m-colouration of MP (D). A previous result is generalized.