# Kernels by monochromatic paths and the color-class digraph

• Volume: 31, Issue: 2, page 273-281
• ISSN: 2083-5892

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## Abstract

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An m-colored digraph is a digraph whose arcs are colored with m colors. A directed path is monochromatic when its arcs are colored alike. A set S ⊆ V(D) is a kernel by monochromatic paths whenever the two following conditions hold: 1. For any x,y ∈ S, x ≠ y, there is no monochromatic directed path between them. 2. For each z ∈ (V(D)-S) there exists a zS-monochromatic directed path. In this paper it is introduced the concept of color-class digraph to prove that if D is an m-colored strongly connected finite digraph such that: (i) Every closed directed walk has an even number of color changes, (ii) Every directed walk starting and ending with the same color has an even number of color changes, then D has a kernel by monochromatic paths. This result generalizes a classical result by Sands, Sauer and Woodrow which asserts that any 2-colored digraph has a kernel by monochromatic paths, in case that the digraph D be a strongly connected digraph.

## How to cite

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Hortensia Galeana-Sánchez. "Kernels by monochromatic paths and the color-class digraph." Discussiones Mathematicae Graph Theory 31.2 (2011): 273-281. <http://eudml.org/doc/271051>.

@article{HortensiaGaleana2011,
abstract = { An m-colored digraph is a digraph whose arcs are colored with m colors. A directed path is monochromatic when its arcs are colored alike. A set S ⊆ V(D) is a kernel by monochromatic paths whenever the two following conditions hold: 1. For any x,y ∈ S, x ≠ y, there is no monochromatic directed path between them. 2. For each z ∈ (V(D)-S) there exists a zS-monochromatic directed path. In this paper it is introduced the concept of color-class digraph to prove that if D is an m-colored strongly connected finite digraph such that: (i) Every closed directed walk has an even number of color changes, (ii) Every directed walk starting and ending with the same color has an even number of color changes, then D has a kernel by monochromatic paths. This result generalizes a classical result by Sands, Sauer and Woodrow which asserts that any 2-colored digraph has a kernel by monochromatic paths, in case that the digraph D be a strongly connected digraph. },
author = {Hortensia Galeana-Sánchez},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {kernel; kernel by monochromatic paths; the color-class digraph; monochromatic paths; color-class digraph},
language = {eng},
number = {2},
pages = {273-281},
title = {Kernels by monochromatic paths and the color-class digraph},
url = {http://eudml.org/doc/271051},
volume = {31},
year = {2011},
}

TY - JOUR
AU - Hortensia Galeana-Sánchez
TI - Kernels by monochromatic paths and the color-class digraph
JO - Discussiones Mathematicae Graph Theory
PY - 2011
VL - 31
IS - 2
SP - 273
EP - 281
AB - An m-colored digraph is a digraph whose arcs are colored with m colors. A directed path is monochromatic when its arcs are colored alike. A set S ⊆ V(D) is a kernel by monochromatic paths whenever the two following conditions hold: 1. For any x,y ∈ S, x ≠ y, there is no monochromatic directed path between them. 2. For each z ∈ (V(D)-S) there exists a zS-monochromatic directed path. In this paper it is introduced the concept of color-class digraph to prove that if D is an m-colored strongly connected finite digraph such that: (i) Every closed directed walk has an even number of color changes, (ii) Every directed walk starting and ending with the same color has an even number of color changes, then D has a kernel by monochromatic paths. This result generalizes a classical result by Sands, Sauer and Woodrow which asserts that any 2-colored digraph has a kernel by monochromatic paths, in case that the digraph D be a strongly connected digraph.
LA - eng
KW - kernel; kernel by monochromatic paths; the color-class digraph; monochromatic paths; color-class digraph
UR - http://eudml.org/doc/271051
ER -

## References

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