ArticleOriginal scientific text
Title
Kernels by monochromatic paths and the color-class digraph
Authors 1
Affiliations
- Instituto de Matemáticas, Universidad Nacional Autónoma de México, Area de la Investigación Cientifica, Ciudad Universitaria, 04510, México, D.F., México
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.
Keywords
kernel, kernel by monochromatic paths, the color-class digraph
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Pages:
273-281
Main language of publication
English
Received
2009-11-24
Accepted
2010-12-02
Published
2011
DOI: 10.7151/dmgt.1544
Exact and natural sciences