# Kernels and cycles' subdivisions in arc-colored tournaments

• Volume: 29, Issue: 1, page 101-117
• ISSN: 2083-5892

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## Abstract

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Let D be a digraph. D is said to be an m-colored digraph if the arcs of D are colored with m colors. A path P in D is called monochromatic if all of its arcs are colored alike. Let D be an m-colored digraph. A set N ⊆ V(D) is said to be a kernel by monochromatic paths of D if it satisfies the following conditions: a) for every pair of different vertices u,v ∈ N there is no monochromatic directed path between them; and b) for every vertex x ∈ V(D)-N there is a vertex n ∈ N such that there is an xn-monochromatic directed path in D. In this paper we prove that if T is an arc-colored tournament which does not contain certain subdivisions of cycles then it possesses a kernel by monochromatic paths. These results generalize a well known sufficient condition for the existence of a kernel by monochromatic paths obtained by Shen Minggang in 1988 and another one obtained by Hahn et al. in 2004. Some open problems are proposed.

## How to cite

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Pietra Delgado-Escalante, and Hortensia Galeana-Sánchez. "Kernels and cycles' subdivisions in arc-colored tournaments." Discussiones Mathematicae Graph Theory 29.1 (2009): 101-117. <http://eudml.org/doc/270273>.

abstract = {Let D be a digraph. D is said to be an m-colored digraph if the arcs of D are colored with m colors. A path P in D is called monochromatic if all of its arcs are colored alike. Let D be an m-colored digraph. A set N ⊆ V(D) is said to be a kernel by monochromatic paths of D if it satisfies the following conditions: a) for every pair of different vertices u,v ∈ N there is no monochromatic directed path between them; and b) for every vertex x ∈ V(D)-N there is a vertex n ∈ N such that there is an xn-monochromatic directed path in D. In this paper we prove that if T is an arc-colored tournament which does not contain certain subdivisions of cycles then it possesses a kernel by monochromatic paths. These results generalize a well known sufficient condition for the existence of a kernel by monochromatic paths obtained by Shen Minggang in 1988 and another one obtained by Hahn et al. in 2004. Some open problems are proposed.},
author = {Pietra Delgado-Escalante, Hortensia Galeana-Sánchez},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {kernel; kernel by monochromatic paths; tournament},
language = {eng},
number = {1},
pages = {101-117},
title = {Kernels and cycles' subdivisions in arc-colored tournaments},
url = {http://eudml.org/doc/270273},
volume = {29},
year = {2009},
}

TY - JOUR
AU - Hortensia Galeana-Sánchez
TI - Kernels and cycles' subdivisions in arc-colored tournaments
JO - Discussiones Mathematicae Graph Theory
PY - 2009
VL - 29
IS - 1
SP - 101
EP - 117
AB - Let D be a digraph. D is said to be an m-colored digraph if the arcs of D are colored with m colors. A path P in D is called monochromatic if all of its arcs are colored alike. Let D be an m-colored digraph. A set N ⊆ V(D) is said to be a kernel by monochromatic paths of D if it satisfies the following conditions: a) for every pair of different vertices u,v ∈ N there is no monochromatic directed path between them; and b) for every vertex x ∈ V(D)-N there is a vertex n ∈ N such that there is an xn-monochromatic directed path in D. In this paper we prove that if T is an arc-colored tournament which does not contain certain subdivisions of cycles then it possesses a kernel by monochromatic paths. These results generalize a well known sufficient condition for the existence of a kernel by monochromatic paths obtained by Shen Minggang in 1988 and another one obtained by Hahn et al. in 2004. Some open problems are proposed.
LA - eng
KW - kernel; kernel by monochromatic paths; tournament
UR - http://eudml.org/doc/270273
ER -

## References

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