On monochromatic paths and monochromatic 4-cycles in edge coloured bipartite tournaments

AbstractWe call the digraph D an m-coloured digraph if the arcs of D are coloured with m colours. A directed path (or a directed cycle) is called monochromatic if all of its arcs are coloured alike.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.(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 it is proved that if D is an m-coloured bipartite tournament such that every directed cycle of length 4 is monochromatic, then D has a kernel by monochromatic paths