# DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

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# Discussiones Mathematicae Graph Theory

## γ-cycles and transitivity by monochromatic paths in arc-coloured digraphs

 Enrique Casas-Bautista(1) Hortensia Galeana-Sánchez(2) and Rocío Rojas-Monroy(1) (1) Facultad de Ciencias, Universidad Autónoma del Estado de México Instituto Literario No. 100, Centro, 50000 Toluca, Edo. de México, México (2) Instituto de Matemáticas Universidad Nacional Autónoma de México Ciudad Universitaria, México, D.F. 04510, México e-mail: ecasasb@uaemex.mx, hgaleana@matem.unam.mx mrrm@uaemex.mx

## Abstract

We call the digraph D an m-coloured digraph if its arcs are coloured with m colours. If D is an m-coloured digraph and a ∈ A(D), colour(a) will denote the colour has been used on a. A path (or a cycle) is called monochromatic if all of its arcs are coloured alike. A γ-cycle in D is a sequence of vertices, say γ = ( u0,u1,...,un), such that ui ≠ uj if i ≠ j and for every i ∈ {0,1,...,n } there is a uiui+1-monochromatic path in D and there is no ui+1ui-monochromatic path in D (the indices of the vertices will be taken mod n+1). A set N ⊆ V(D) is said to be a kernel by monochromatic paths if it satisfies the following two conditions: (i) for every pair of different vertices u,v ∈ N there is no monochromatic path between them and; (ii) for every vertex x ∈ V(D)\N there is a vertex y ∈ N such that there is an xy-monochromatic path.

Let D be a finite m-coloured digraph. Suppose that { C1, C2 } is a partition of C, the set of colours of D, and Di will be the spanning subdigraph of D such that A(Di) = { a ∈ A(D)   | colour(a) ∈ Ci }. In this paper, we give some sufficient conditions for the existence of a kernel by monochromatic paths in a digraph with the structure mentioned above. In particular we obtain an extension of the original result by B. Sands, N. Sauer and R. Woodrow that asserts: Every 2-coloured digraph has a kernel by monochromatic paths. Also, we extend other results obtained before where it is proved that under some conditions an m-coloured digraph has no γ-cycles.

Keywords: digraph, kernel, kernel by monochromatic paths, γ-cycle

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

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