Monochromatic paths and monochromatic cycles in edge-coloured k-partite tournaments

Abstract

We call the digraph D an m-coloured digraph if the arcs of D are coloured with m colours. A subdigraph H of D is called monochromatic if all of its arcs are coloured alike. A set N subset of 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. (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 it is proved that if D is an m-coloured k-partite tournament such that every directed cycle of length 3 and every directed cycle of length 4 is monochromatic, then D has a kernel by monochromatic paths. Some previous results are generalized.