- 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