Kernels in edge-colored digraphs

Abstract 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 directed cycle is called quasi-monochromatic if with at most one exception all of its arcs are coloured alike. A set N C 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 every vertex x ∈ V ( D )− N there is a vertex y ∈ N such that there is an xy -monochromatic directed path. In this paper I survey sufficient conditions for a m -coloured digraph to have a kernel by monochromatic paths. I also prove that if D is an m -coloured digraph resulting from the deletion of a single arc of some m -coloured tournament and every directed cycle of length at most 4 is quasi-monochromatic then D has a kernel by monochromatic paths.