Kernels by monochromatic paths and the color-class digraph
Discussiones Mathematicae Graph Theory, Tome 31 (2011) no. 2, p. 273.

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An m-colored digraph is a digraph whose arcs are colored with m colors. A directed path is monochromatic when its arcs are colored alike. A set S ⊆ V(D) is a kernel by monochromatic paths whenever the two following conditions hold: 1. For any x,y ∈ S, x ≠ y, there is no monochromatic directed path between them. 2. For each z ∈ (V(D)-S) there exists a zS-monochromatic directed path. In this paper it is introduced the concept of color-class digraph to prove that if D is an m-colored strongly connected finite digraph such that: (i) Every closed directed walk has an even number of color changes, (ii) Every directed walk starting and ending with the same color has an even number of color changes, then D has a kernel by monochromatic paths. This result generalizes a classical result by Sands, Sauer and Woodrow which asserts that any 2-colored digraph has a kernel by monochromatic paths, in case that the digraph D be a strongly connected digraph.
Classification : 05C20
Mots-clés : kernel, kernel by monochromatic paths, the color-class digraph, monochromatic paths, color-class digraph 
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Hortensia Galeana-Sánchez. Kernels by monochromatic paths and the color-class digraph. Discussiones Mathematicae Graph Theory, Tome 31 (2011) no. 2, p. 273. https://geodesic-test.mathdoc.fr/item/DMGT_2011__31_2_271051/