Discussiones Mathematicae Graph Theory 31(4) (2011) 791-820
doi: 10.7151/dmgt.1580

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On monochromatic paths and bicolored subdigraphs in arc-colored tournaments

Pietra Delgado-Escalante
and
Hortensia Galeana-Sánchez

Instituto de Matemáticas
U.N.A.M. 'Area de la investigación cient'ifica
Circuito Exterior, Ciudad Universitaria
Coyoacán 04510, México, D.F. México

Abstract

Consider an arc-colored digraph. A set of vertices N is a kernel by monochromatic paths if all pairs of distinct vertices of N have no monochromatic directed path between them and if for every vertex v not in N there exists n ∈ N such that there is a monochromatic directed path from v to n.

In this paper we prove different sufficient conditions which imply that an arc-colored tournament has a kernel by monochromatic paths. Our conditions concerns to some subdigraphs of T and its quasimonochromatic and bicolor coloration. We also prove that our conditions are not mutually implied and that they are not implied by those known previously. Besides some open problems are proposed.

Keywords: kernel, kernel by monochromatic paths, tournament

2010 Mathematics Subject Classification: 05C20, 05C38, 05C69.

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Received 20 September 2007
Revised 8 December 2010
Accepted 8 December 2010