Discussiones Mathematicae Graph Theory 31(2) (2011) 273-281
doi: 10.7151/dmgt.1544

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Kernels by monochromatic paths
and the color-class digraph

Hortensia Galeana-Sánchez

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

2010 Mathematics Subject Classification: 05C20.

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Received 24 November 2009
Revised 2 December 2010
Accepted 27 January 2011