We call the digraph D an m-colored digraph if the arcs of D are colored with m colors. A path (or a cycle) is called monochromatic if all of its arcs are colored alike. A cycle is called a quasi-monochromatic cycle if with at most one exception all of its arcs are colored alike. A subdigraph H in D is called rainbow if all its arcs have different colors. 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. The closure of D, denoted by ℭ(D), is the m-colored multidigraph defined as follows: V(ℭ(D)) = V(D), A(ℭ(D)) = A(D)∪(u,v) with color i | there exists a uv-monochromatic path colored i contained in D. Notice that for any digraph D, ℭ (ℭ(D)) ≅ ℭ(D) and D has a kernel by monochromatic paths if and only if ℭ(D) has a kernel. Let D be a finite m-colored digraph. Suppose that there is a partition C = C₁ ∪ C₂ of the set of colors of D such that every cycle in the subdigraph spanned by the arcs with colors in is monochromatic. We show that if ℭ(D) does not contain neither rainbow triangles nor rainbow P₃ involving colors of both C₁ and C₂, then D has a kernel by monochromatic paths. This result is a wide extension of the original result by Sands, Sauer and Woodrow that asserts: Every 2-colored digraph has a kernel by monochromatic paths (since in this case there are no rainbow triangles in ℭ(D)).
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1545,
author = {Hortensia Galeana-S\'anchez and Guadalupe Gayt\'an-G\'omez and Roc\'\i o Rojas-Monroy},
title = {Monochromatic cycles and monochromatic paths in arc-colored digraphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {31},
year = {2011},
pages = {283-292},
zbl = {1234.05112},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1545}
}
Hortensia Galeana-Sánchez; Guadalupe Gaytán-Gómez; Rocío Rojas-Monroy. Monochromatic cycles and monochromatic paths in arc-colored digraphs. Discussiones Mathematicae Graph Theory, Tome 31 (2011) pp. 283-292. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1545/
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