- Tytuł:
- Monochromatic cycles and monochromatic paths in arc-colored digraphs
- Autorzy:
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Galeana-Sánchez, Hortensia
Gaytán-Gómez, Guadalupe
Rojas-Monroy, Rocío - Powiązania:
- https://bibliotekanauki.pl/articles/743875.pdf
- Data publikacji:
- 2011
- Wydawca:
- Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
- Tematy:
-
kernel
kernel by monochromatic paths
monochromatic cycles - Opis:
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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 $D[C_i]$ spanned by the arcs with colors in $C_i$ 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)). - Źródło:
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Discussiones Mathematicae Graph Theory; 2011, 31, 2; 283-292
2083-5892 - Pojawia się w:
- Discussiones Mathematicae Graph Theory
- Dostawca treści:
- Biblioteka Nauki
Artykuł