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.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1544,
author = {Hortensia Galeana-S\'anchez},
title = {Kernels by monochromatic paths and the color-class digraph},
journal = {Discussiones Mathematicae Graph Theory},
volume = {31},
year = {2011},
pages = {273-281},
zbl = {1234.05111},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1544}
}
Hortensia Galeana-Sánchez. Kernels by monochromatic paths and the color-class digraph. Discussiones Mathematicae Graph Theory, Tome 31 (2011) pp. 273-281. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1544/
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