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Wyświetlanie 1-4 z 4
Tytuł:
Monochromatic paths and quasi-monochromatic cycles in edge-coloured bipartite tournaments
Autorzy:
Galeana-Sanchez, Hortensia
Rojas-Monroy, Rocío
Powiązania:
https://bibliotekanauki.pl/articles/743320.pdf
Data publikacji:
2008
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
kernel
kernel by monochromatic paths
bipartite tournament
Opis:
We call the digraph D an m-coloured digraph if the arcs of D are coloured with m colours. A directed path (or a directed cycle) is called monochromatic if all of its arcs are coloured alike. A directed cycle is called quasi-monochromatic if with at most one exception all of its arcs are coloured alike. 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 directed 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 directed path.
In this paper it is proved that if D is an m-coloured bipartite tournament such that: every directed cycle of length 4 is quasi-monochromatic, every directed cycle of length 6 is monochromatic, and D has no induced particular 6-element bipartite tournament T̃₆, then D has a kernel by monochromatic paths.
Źródło:
Discussiones Mathematicae Graph Theory; 2008, 28, 2; 285-306
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
γ-Cycles And Transitivity By Monochromatic Paths In Arc-Coloured Digraphs
Autorzy:
Casas-Bautista, Enrique
Galeana-Sánchez, Hortensia
Rojas-Monroy, Rocío
Powiązania:
https://bibliotekanauki.pl/articles/30146505.pdf
Data publikacji:
2013-07-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
digraph
kernel
kernel by monochromatic paths
γ-cycle
Opis:
We call the digraph D an m-coloured digraph if its arcs are coloured with m colours. If D is an m-coloured digraph and a ∈ A(D), colour(a) will denote the colour has been used on a. A path (or a cycle) is called monochromatic if all of its arcs are coloured alike. A γ-cycle in D is a sequence of vertices, say γ = (u0, u1, . . ., un), such that ui ≠ uj if i ≠ j and for every i ∈ {0, 1, . . ., n} there is a uiui+1-monochromatic path in D and there is no ui+1ui-monochromatic path in D (the indices of the vertices will be taken mod n+1). 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. Let D be a finite m-coloured digraph. Suppose that {C1,C2} is a partition of C, the set of colours of D, and Di will be the spanning subdigraph of D such that A(Di) = {a ∈ A(D) | colour(a) ∈ Ci}. In this paper, we give some sufficient conditions for the existence of a kernel by monochromatic paths in a digraph with the structure mentioned above. In particular we obtain an extension of the original result by B. Sands, N. Sauer and R. Woodrow that asserts: Every 2-coloured digraph has a kernel by monochromatic paths. Also, we extend other results obtained before where it is proved that under some conditions an m-coloured digraph has no γ-cycles.
Źródło:
Discussiones Mathematicae Graph Theory; 2013, 33, 3; 493-507
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
$ \gamma $-Cycles In Arc-Colored Digraphs
Autorzy:
Galeana-Sánchez, Hortensia
Gaytán-Gómez, Guadalupe
Rojas-Monroy, Rocío
Powiązania:
https://bibliotekanauki.pl/articles/31341163.pdf
Data publikacji:
2016-02-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
digraph
kernel
kernel by monochromatic paths
γ-cycle
Opis:
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 \subseteq V (D) $ is said to be a kernel by monochromatic paths of $D$ if it satisfies the two following conditions: (i) for every pair of different vertices $ u, v \in N $ there is no monochromatic path in $D$ between them, and (ii) for every vertex $ x \in V (D) − N $ there is a vertex $ y \in N $ such that there is an $xy$-monochromatic path in $D$. A $\gamma$-cycle in $D$ is a sequence of different vertices $ \gamma = (u_0, u_1, . . ., u_n, u_0)$ such that for every $ i \in {0, 1, . . ., n}$: (i) there is a $u_i u_{i+1}$-monochromatic path, and (ii) there is no $u_{i+1}u_i$-monochromatic path. The addition over the indices of the vertices of $ \gamma $ is taken modulo $(n + 1)$. If $D$ is an $m$-colored digraph, then the closure of $D$, denoted by $ \mathfrak{C}(D)$, is the $m$-colored multidigraph defined as follows: $ V (\mathfrak{C} (D)) = V (D) $, $ A( \mathfrak{C} (D)) = A(D) \cup \{ (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 = C_1 \cup C_2 $ of the set of colors of $D$ such that: (1) $ D[ \hat{C}_i ] $ (the subdigraph spanned by the arcs with colors in $ C_i) $ contains no $ \gamma $-cycles for $ i \in {1, 2} $; (2) If $ \mathfrak{C}(D) $ contains a rainbow $ C_3 = (x_0, z, w, x_0) $ involving colors of $ C_1 $ and $ C_2 $, then $ (x_0, w) \in A(\mathfrak{C} (D)) $ or $ (z, x_0) \in A( \mathfrak{C} (D)) $; (3) If $ \mathfrak{C}(D) $ contains a rainbow $ P_3 = (u, z, w, x_0) $ involving colors of $ C_1 $ and $ C_2 $, then at least one of the following pairs of vertices is an arc in $ \mathfrak{C} (D) $: $ (u, w) $, $ (w, u) $, $ (x_0, u) $, $ (u, x_0) $, $ (x_0, w) $, $ (z, u) $, $ (z, x_0) $. Then $D$ has a kernel by monochromatic paths. This theorem can be applied to all those digraphs that contain no $ \gamma $-cycles. Generalizations of many previous results are obtained as a direct consequence of this theorem.
Źródło:
Discussiones Mathematicae Graph Theory; 2016, 36, 1; 103-116
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Monochromatic cycles and monochromatic paths in arc-colored digraphs
Autorzy:
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:
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:
Discussiones Mathematicae Graph Theory; 2011, 31, 2; 283-292
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
    Wyświetlanie 1-4 z 4

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