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.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1292,
author = {Hortensia Galeana-S\'anchez and Laura Pastrana Ram\'\i rez and Hugo Alberto Rinc\'on Mej\'\i a},
title = {Kernels in monochromatic path digraphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {25},
year = {2005},
pages = {407-417},
zbl = {1104.05030},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1292}
}
Hortensia Galeana-Sánchez; Laura Pastrana Ramírez; Hugo Alberto Rincón Mejía. Kernels in monochromatic path digraphs. Discussiones Mathematicae Graph Theory, Tome 25 (2005) pp. 407-417. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1292/
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