# Kernels in monochromatic path digraphs

• Volume: 25, Issue: 3, page 407-417
• ISSN: 2083-5892

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## Abstract

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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.

## How to cite

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Hortensia Galeana-Sánchez, Laura Pastrana Ramírez, and Hugo Alberto Rincón Mejía. "Kernels in monochromatic path digraphs." Discussiones Mathematicae Graph Theory 25.3 (2005): 407-417. <http://eudml.org/doc/270464>.

@article{HortensiaGaleana2005,
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. },
author = {Hortensia Galeana-Sánchez, Laura Pastrana Ramírez, Hugo Alberto Rincón Mejía},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {kernel; line digraph; kernel by monochromatic paths; monochromatic path digraph; edge-coloured digraph; line graph; edge-colored digraph},
language = {eng},
number = {3},
pages = {407-417},
title = {Kernels in monochromatic path digraphs},
url = {http://eudml.org/doc/270464},
volume = {25},
year = {2005},
}

TY - JOUR
AU - Hortensia Galeana-Sánchez
AU - Laura Pastrana Ramírez
AU - Hugo Alberto Rincón Mejía
TI - Kernels in monochromatic path digraphs
JO - Discussiones Mathematicae Graph Theory
PY - 2005
VL - 25
IS - 3
SP - 407
EP - 417
AB - 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.
LA - eng
KW - kernel; line digraph; kernel by monochromatic paths; monochromatic path digraph; edge-coloured digraph; line graph; edge-colored digraph
UR - http://eudml.org/doc/270464
ER -

## References

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