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γ-Cycles In Arc-Colored Digraphs

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Galeana-Sánchez H., Gaytán-Gómez G., Rojas-Monroy R.

Discussiones Mathematicae Graph Theory

2016

tom 36

nr 1

103-116

We call a digraph D an m-colored digraph if the arcs of D are colored with m colors. A directed path (or a directed cycle) is called monochromatic if all of its arcs are colored alike. A subdigraph H in D is called rainbow if all of its arcs have different colors. A set N ⊆ V (D) is said to be a kernel by monochromatic paths of D if it satisfies the two following conditions: for every pair of different vertices u, v ∈ N there is no monochromatic path in D between them, and for every vertex x ∈ V (D) − N there is a vertex y ∈ N such that there is an xy-monochromatic path in D. A γ-cycle in D is a sequence of different vertices γ = (u0, u1, . . . , un, u0) such that for every i ∈ {0, 1, . . . , n}: there is a uiui+1-monochromatic path, and there is no ui+1ui-monochromatic path. The addition over the indices of the vertices of γ is taken modulo (n + 1). If D is an m-colored digraph, then 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}. In this work, we prove the following result. Let D be a finite m-colored digraph which satisfies that there is a partition C = C1 ∪ C2 of the set of colors of D such that: D[Ĉi] (the subdigraph spanned by the arcs with colors in Ci) contains no γ-cycles for i ∈ {1, 2}; If ℭ (D) contains a rainbow C3 = (x0, z, w, x0) involving colors of C1 and C2, then (x0, w) ∈ A(ℭ (D)) or (z, x0) ∈ A(ℭ (D)); If ℭ (D) contains a rainbow P3 = (u, z, w, x0) involving colors of C1 and C2, then at least one of the following pairs of vertices is an arc in ℭ (D): (u, w), (w, u), (x0, u), (u, x0), (x0, w), (z, u), (z, x0). Then D has a kernel by monochromatic paths. This theorem can be applied to all those digraphs that contain no γ-cycles. Generalizations of many previous results are obtained as a direct consequence of this theorem.

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Monochromatic cycles and monochromatic paths in arc-colored digraphs

100%

Galeana-Sánchez H., Gaytán-Gómez G., Rojas-Monroy R.

Discussiones Mathematicae Graph Theory

2011

tom 31

nr 2

283-292

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)).

Ograniczanie wyników

2 Discussiones Mathematicae Graph Theory

2 Galeana-Sánchez H.

2 Gaytán-Gómez G.

2 Rojas-Monroy R.

1 2016

1 2011

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γ-Cycles In Arc-Colored Digraphs

Monochromatic cycles and monochromatic paths in arc-colored digraphs