Geodesic
Monochromatic paths and monochromatic sets of arcs in 3-quasitransitive digraphs
Discussiones Mathematicae. Graph Theory, Tome 29 (2009) no. 2, pp. 337-347

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We call the digraph D an m-coloured digraph if the arcs of D are coloured with m colours. A directed path is called monochromatic if all of its arcs are coloured alike. A set N of vertices of D is called a kernel by monochromatic paths if for every pair of vertices of N there is no monochromatic path between them and for every vertex v ∉ N there is a monochromatic path from v to N. We denote by A⁺(u) the set of arcs of D that have u as the initial vertex. We prove that if D is an m-coloured 3-quasitransitive digraph such that for every vertex u of D, A⁺(u) is monochromatic and D satisfies some colouring conditions over one subdigraph of D of order 3 and two subdigraphs of D of order 4, then D has a kernel by monochromatic paths.
Keywords: m-coloured digraph, 3-quasitransitive digraph, kernel by monochromatic paths, γ-cycle, quasi-monochromatic digraph 
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Galeana-Sánchez, Hortensia; Rojas-Monroy, R.; Zavala, B. Monochromatic paths and monochromatic sets of arcs in 3-quasitransitive digraphs. Discussiones Mathematicae. Graph Theory, Tome 29 (2009) no. 2, pp. 337-347. http://geodesic.mathdoc.fr/item/DMGT_2009_29_2_a8/