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 \(N \subseteq V(D)\) is said to be a kernel by monochromatic paths if it satisfies the following two conditions:
(i) For every pair of different vertices \(u,v \in N\) there is no monochromatic directed path between them.
(ii) For every vertex \(x \in V(D) – N\), there is a vertex \(y \in N\) 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.
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