Kernels in edge-colored digraphs

We call the digraph D an m-coloured digraph if the arcs of D are coloured with rn 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 subset of or equal to 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 is an element of N there is no monochromatic directed path between them and; (ii) for every vertex x is an element of V(D)-N there is a vertex y is an element of 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 are 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. (C) 1998 Elsevier Science B.V. All rights reserved.