# γ-Cycles And Transitivity By Monochromatic Paths In Arc-Coloured Digraphs

• Volume: 33, Issue: 3, page 493-507
• 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. If D is an m-coloured digraph and a ∈ A(D), colour(a) will denote the colour has been used on a. A path (or a cycle) is called monochromatic if all of its arcs are coloured alike. A γ-cycle in D is a sequence of vertices, say γ = (u0, u1, . . . , un), such that ui ≠ uj if i ≠ j and for every i ∈ 0, 1, . . . , n there is a uiui+1-monochromatic path in D and there is no ui+1ui-monochromatic path in D (the indices of the vertices will be taken mod n+1). 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 path between them and; (ii) for every vertex x ∈ V (D) N there is a vertex y ∈ N such that there is an xy-monochromatic path. Let D be a finite m-coloured digraph. Suppose that C1,C2 is a partition of C, the set of colours of D, and Di will be the spanning subdigraph of D such that A(Di) = a ∈ A(D) | colour(a) ∈ Ci. In this paper, we give some sufficient conditions for the existence of a kernel by monochromatic paths in a digraph with the structure mentioned above. In particular we obtain an extension of the original result by B. Sands, N. Sauer and R. Woodrow that asserts: Every 2-coloured digraph has a kernel by monochromatic paths. Also, we extend other results obtained before where it is proved that under some conditions an m-coloured digraph has no γ-cycles.

## How to cite

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Enrique Casas-Bautista, Hortensia Galeana-Sánchez, and Rocío Rojas-Monroy. "γ-Cycles And Transitivity By Monochromatic Paths In Arc-Coloured Digraphs." Discussiones Mathematicae Graph Theory 33.3 (2013): 493-507. <http://eudml.org/doc/268248>.

@article{EnriqueCasas2013,
abstract = {We call the digraph D an m-coloured digraph if its arcs are coloured with m colours. If D is an m-coloured digraph and a ∈ A(D), colour(a) will denote the colour has been used on a. A path (or a cycle) is called monochromatic if all of its arcs are coloured alike. A γ-cycle in D is a sequence of vertices, say γ = (u0, u1, . . . , un), such that ui ≠ uj if i ≠ j and for every i ∈ 0, 1, . . . , n there is a uiui+1-monochromatic path in D and there is no ui+1ui-monochromatic path in D (the indices of the vertices will be taken mod n+1). 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 path between them and; (ii) for every vertex x ∈ V (D) N there is a vertex y ∈ N such that there is an xy-monochromatic path. Let D be a finite m-coloured digraph. Suppose that C1,C2 is a partition of C, the set of colours of D, and Di will be the spanning subdigraph of D such that A(Di) = a ∈ A(D) | colour(a) ∈ Ci. In this paper, we give some sufficient conditions for the existence of a kernel by monochromatic paths in a digraph with the structure mentioned above. In particular we obtain an extension of the original result by B. Sands, N. Sauer and R. Woodrow that asserts: Every 2-coloured digraph has a kernel by monochromatic paths. Also, we extend other results obtained before where it is proved that under some conditions an m-coloured digraph has no γ-cycles.},
author = {Enrique Casas-Bautista, Hortensia Galeana-Sánchez, Rocío Rojas-Monroy},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {digraph; kernel; kernel by monochromatic paths; γ-cycle; -cycle},
language = {eng},
number = {3},
pages = {493-507},
title = {γ-Cycles And Transitivity By Monochromatic Paths In Arc-Coloured Digraphs},
url = {http://eudml.org/doc/268248},
volume = {33},
year = {2013},
}

TY - JOUR
AU - Enrique Casas-Bautista
AU - Hortensia Galeana-Sánchez
AU - Rocío Rojas-Monroy
TI - γ-Cycles And Transitivity By Monochromatic Paths In Arc-Coloured Digraphs
JO - Discussiones Mathematicae Graph Theory
PY - 2013
VL - 33
IS - 3
SP - 493
EP - 507
AB - We call the digraph D an m-coloured digraph if its arcs are coloured with m colours. If D is an m-coloured digraph and a ∈ A(D), colour(a) will denote the colour has been used on a. A path (or a cycle) is called monochromatic if all of its arcs are coloured alike. A γ-cycle in D is a sequence of vertices, say γ = (u0, u1, . . . , un), such that ui ≠ uj if i ≠ j and for every i ∈ 0, 1, . . . , n there is a uiui+1-monochromatic path in D and there is no ui+1ui-monochromatic path in D (the indices of the vertices will be taken mod n+1). 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 path between them and; (ii) for every vertex x ∈ V (D) N there is a vertex y ∈ N such that there is an xy-monochromatic path. Let D be a finite m-coloured digraph. Suppose that C1,C2 is a partition of C, the set of colours of D, and Di will be the spanning subdigraph of D such that A(Di) = a ∈ A(D) | colour(a) ∈ Ci. In this paper, we give some sufficient conditions for the existence of a kernel by monochromatic paths in a digraph with the structure mentioned above. In particular we obtain an extension of the original result by B. Sands, N. Sauer and R. Woodrow that asserts: Every 2-coloured digraph has a kernel by monochromatic paths. Also, we extend other results obtained before where it is proved that under some conditions an m-coloured digraph has no γ-cycles.
LA - eng
KW - digraph; kernel; kernel by monochromatic paths; γ-cycle; -cycle
UR - http://eudml.org/doc/268248
ER -

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