DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

https://doi.org/10.7151/dmgt

Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory 25(3) (2005) 407-417
DOI: 10.7151/dmgt.1292

KERNELS IN MONOCHROMATIC PATH DIGRAPHS

Hortensia Galeana-Sánchez

Instituto de Matemáticas, UNAM
Universidad Nacional Autónoma de México
Ciudad Universitaria
04510, México, D.F. MÉXICO
e-mail: hgaleana@matem.unam.mx

Laura Pastrana Ramírez and Hugo Alberto Rincón Mejía

Departamento de Matemáticas
Facultad de Ciencias
Universidad Nacional Autónoma de México
Ciudad Universitaria
04510, México, D.F. MÉXICO

Abstract

We call the digraph D an m-coloured digraph if its arcs are coloured with m colours. A directed path (or a directed cycle) is called monochromatic if all of its arcs are coloured alike.

Let D be an m-coloured digraph. 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 each vertex x ∈ (V(D)−N) there is a vertex y ∈ N such that there is an xy-monochromatic directed path.

In this paper is defined the monochromatic path digraph of D, MP(D), and the inner m-colouration of MP(D). Also it is proved that if D is an m-coloured digraph without monochromatic directed cycles, then the number of kernels by monochromatic paths in D is equal to the number of kernels by monochromatic paths in the inner m-colouration of MP(D). A previous result is generalized.

Keywords: kernel, line digraph, kernel by monochromatic paths, monochromatic path digraph, edge-coloured digraph.

2000 Mathematic Subject Classification: 05C20.

References

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Received 3 August 2004
Revised 10 December 2004