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.
@article{bwmeta1.element.doi-10_7151_dmgt_1695,
author = {Enrique Casas-Bautista and Hortensia Galeana-S\'anchez and Roc\'\i o Rojas-Monroy},
title = {$\gamma$-Cycles And Transitivity By Monochromatic Paths In Arc-Coloured Digraphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {33},
year = {2013},
pages = {493-507},
zbl = {1275.05025},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_7151_dmgt_1695}
}
Enrique Casas-Bautista; Hortensia Galeana-Sánchez; Rocío Rojas-Monroy. γ-Cycles And Transitivity By Monochromatic Paths In Arc-Coloured Digraphs. Discussiones Mathematicae Graph Theory, Tome 33 (2013) pp. 493-507. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1695/
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