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

We 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 satis5es the following two conditions: (i) For every pair of di7erent 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. c © 2004 Elsevier B.V. All rights reserved.