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Discussiones Mathematicae Graph Theory 29(2) (2009) 337-347
DOI: https://doi.org/10.7151/dmgt.1450

MONOCHROMATIC PATHS AND MONOCHROMATIC SETS OF ARCS IN 3-QUASITRANSITIVE DIGRAPHS

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. 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 of N there is no monochromatic path between them and for every vertex v ∉ N there is a monochromatic path from v to N. We denote by A⁺(u) the set of arcs of D that have u as the initial vertex. We prove that if D is an m-coloured 3-quasitransitive digraph such that for every vertex u of D, A⁺(u) is monochromatic and D satisfies some colouring conditions over one subdigraph of D of order 3 and two subdigraphs of D of order 4, then D has a kernel by monochromatic paths.

Keywords: m-coloured digraph, 3-quasitransitive digraph, kernel by monochromatic paths, γ-cycle, quasi-monochromatic digraph.

2000 Mathematics Subject Classification: 05C15, 05C20.

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Received 6 November 2007
Revised 26 February 2009
Accepted 27 February 2009