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

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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