# Monochromatic paths and monochromatic sets of arcs in quasi-transitive digraphs

• Volume: 30, Issue: 4, page 545-553
• ISSN: 2083-5892

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## Abstract

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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.

## How to cite

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Hortensia Galeana-Sánchez, R. Rojas-Monroy, and B. Zavala. "Monochromatic paths and monochromatic sets of arcs in quasi-transitive digraphs." Discussiones Mathematicae Graph Theory 30.4 (2010): 545-553. <http://eudml.org/doc/270827>.

@article{HortensiaGaleana2010,
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.},
author = {Hortensia Galeana-Sánchez, R. Rojas-Monroy, B. Zavala},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {m-coloured quasi-transitive digraph; kernel by monochromatic paths; -coloured quasi-transitive digraph},
language = {eng},
number = {4},
pages = {545-553},
title = {Monochromatic paths and monochromatic sets of arcs in quasi-transitive digraphs},
url = {http://eudml.org/doc/270827},
volume = {30},
year = {2010},
}

TY - JOUR
AU - Hortensia Galeana-Sánchez
AU - R. Rojas-Monroy
AU - B. Zavala
TI - Monochromatic paths and monochromatic sets of arcs in quasi-transitive digraphs
JO - Discussiones Mathematicae Graph Theory
PY - 2010
VL - 30
IS - 4
SP - 545
EP - 553
AB - 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.
LA - eng
KW - m-coloured quasi-transitive digraph; kernel by monochromatic paths; -coloured quasi-transitive digraph
UR - http://eudml.org/doc/270827
ER -

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