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Discussiones Mathematicae Graph Theory 30(4) (2010) 545-553
DOI: https://doi.org/10.7151/dmgt.1512

MONOCHROMATIC PATHS AND MONOCHROMATIC SETS OF ARCS IN QUASI-TRANSITIVE 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

Let D be a digraph, V(D) and A(D) will denote the sets of vertices and arcs of D, respectively. We call the digraph D an m-coloured digraph if each arc of D is coloured by an element of {1,2,...,m} where m ≥ 1. 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 there is no monochromatic path between two vertices of N and if for every vertex v not in N there is a monochromatic path from v to some vertex in N. A digraph D is called a quasi-transitive digraph if (u,v) ∈ A(D) and (v,w) ∈ A(D) implies (u,w) ∈ A(D) or (w,u) ∈ A(D). We prove that if D is an m-coloured quasi-transitive digraph such that for every vertex u of D the set of arcs that have u as initial end point is monochromatic and D contains no C₃ (the 3-coloured directed cycle of length 3), then D has a kernel by monochromatic paths.

Keywords: m-coloured quasi-transitive digraph, kernel by monochromatic paths.

2010 Mathematics Subject Classification: 05C15, 05C20.

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Received 21 May 2007
Revised 22 October 2009
Accepted 27 October 2009