Discussiones Mathematicae Graph Theory 29(2) (2009) 349-360
doi: 10.7151/dmgt.1451

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Hortensia Galeana-Sánchez1, R. Rojas-Monroy2 and B. Zavala1

1Instituto de Matemáticas
Universidad Nacional Autónoma de México
Ciudad Universitaria, México, D.F. 04510, México

2Facultad de Ciencias
Universidad Autónoma del Estado de México
Instituto Literario, Centro 50000, Toluca, Edo. de México, México


We call the digraph D an m-coloured digraph if the arcs of D are coloured with m colours and all of them are used. A directed path is called monochromatic if all of its arcs are coloured alike. A set N of vertices of D is called a kernel by monochromatic paths if for every pair of vertices there is no monochromatic path between them and for every vertex v in V(D)∖N there is a monochromatic path from v to some vertex in N. We denote by A+(u) the set of arcs of D that have u as the initial endpoint.

In this paper we introduce the concept of semikernel modulo i by monochromatic paths of an m-coloured digraph. This concept allow us to find sufficient conditions for the existence of a kernel by monochromatic paths in an m-coloured digraph. In particular we deal with bipartite tournaments such that A+(z) is monochromatic for each z ∈ V(D).

Keywords: m-coloured bipartite tournaments, kernel by monochromatic paths, semikernel of D modulo i by monochromatic paths.

2000 Mathematics Subject Classification: 05C15, 05C20.


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Received 6 November 2007
Revised 20 March 2009
Accepted 20 March 2009