Discussiones Mathematicae Graph Theory 29(1) (2009) 101-117
doi: 10.7151/dmgt.1435

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KERNELS AND CYCLES' SUBDIVISIONS IN ARC-COLORED TOURNAMENTS

Pietra Delgado-Escalante and Hortensia Galeana-Sánchez

Instituto de Matemáticas
U.N.A.M. Área de la investigación científica
Circuito Exterior, Ciudad Universitaria
Coyoacán 04510, México, D.F. México
e-mail: pietra@matem.unam.mx
e-mail: hgaleana@matem.unam.mx

Abstract

Let D be a digraph. D is said to be an m-colored digraph if the arcs of D are colored with m colors. A path P in D is called monochromatic if all of its arcs are colored alike. Let D be an m-colored digraph. A set N ⊆ V(D) is said to be a kernel by monochromatic paths of D if it satisfies the following conditions: a) for every pair of different vertices u,v ∈ N there is no monochromatic directed path between them; and b) for every vertex x ∈ V(D)−N there is a vertex n ∈ N such that there is an xn-monochromatic directed path in D.

In this paper we prove that if T is an arc-colored tournament which does not contain certain subdivisions of cycles then it possesses a kernel by monochromatic paths. These results generalize a well known sufficient condition for the existence of a kernel by monochromatic paths obtained by Shen Minggang in 1988 and another one obtained by Hahn et al. in 2004. Some open problems are proposed.

Keywords: kernel, kernel by monochromatic paths, tournament.

2000 Mathematics Subject Classification: 05C20, 05C38, 05C69.

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Received 30 January 2007
Revised 3 December 2008
Accepted 3 December 2008