We call the digraph D an orientation of a graph G if D is obtained from G by the orientation of each edge of G in exactly one of the two possible directions. The digraph D is an m-coloured digraph if the arcs of D are coloured with m-colours.
Let D be an m-coloured digraph. A directed path (or a directed cycle) is called monochromatic if all of its arcs are coloured alike.
A set is said to be a kernel by monochromatic paths if it satisfies the two following conditions: (i) for every pair of different vertices there is no monochromatic directed path between them and (ii) for every vertex there is a vertex such that there is an xy-monochromatic directed path.
In this paper we obtain sufficient conditions for an m-coloured orientation of a graph obtained from by deletion of the arcs of to have a kernel by monochromatic.