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

MONOCHROMATIC PATHS AND MONOCHROMATIC SETS OF ARCS IN BIPARTITE TOURNAMENTS

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

Abstract

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