DMGT

Discussiones Mathematicae Graph Theory 28(2) (2008) 285-306
DOI: https://doi.org/10.7151/dmgt.1406

MONOCHROMATIC PATHS AND QUASI-MONOCHROMATIC CYCLES IN EDGE-COLOURED BIPARTITE TOURNAMENTS

Hortensia Galeana-Sanchez

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

Rocío Rojas-Monroy

Facultad de Ciencias
Universidad Autónoma del Estado de México
Instituto Literario No. 100, 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. A directed path (or a directed cycle) is called monochromatic if all of its arcs are coloured alike. A directed cycle is called quasi-monochromatic if with at most one exception all of its arcs are coloured alike.

A set N ⊆ 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 ∈ N there is no monochromatic directed path between them and
(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 quasi-monochromatic, every directed cycle of length 6 is monochromatic, and D has no induced particular 6-element bipartite tournament T^˜₆, then D has a kernel by monochromatic paths.

Keywords: kernel, kernel by monochromatic paths, bipartite tournament.

2000 Mathematics Subject Classification: 05C20.

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Received 6 July 2007
Revised 10 April 2008
Accepted 10 April 2008