# Kernels by monochromatic paths and the color-class digraph

Discussiones Mathematicae Graph Theory (2011)

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

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topHortensia 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 -

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