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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:
Alternating-Pancyclism in 2-Edge-Colored Graphs
Autorzy:
Cordero-Michel, Narda
Galeana-Sánchez, Hortensia
Powiązania:
https://bibliotekanauki.pl/articles/32222696.pdf
Data publikacji:
2021-08-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
2-edge-colored graph
alternating cycle
alternating-pancyclic graph
Opis:
An alternating cycle in a 2-edge-colored graph is a cycle such that any two consecutive edges have different colors. Let $ G_1, . . ., G_k $ be a collection of pairwise vertex disjoint 2-edge-colored graphs. The colored generalized sum of $ G_1, . . ., G_k $, denoted by $ \oplus_{i=1}^k G_i $, is the set of all 2-edge-colored graphs $G$ such that: (i) \( V(G)= \bigcup _{i=1}^k V(G_i) \), (ii) $ G \langle V(G_i) \rangle \cong G_i $ for $ i = 1, . . ., k $ where $ G \langle V(G_i) \rangle $ has the same coloring as $ G_i $ and (iii) between each pair of vertices in different summands of $G$ there is exactly one edge, with an arbitrary but fixed color. A graph $G$ in $\oplus_{i=1}^k G_i $ will be called a colored generalized sum (c.g.s.) and we will say that $ e \in E(G) $ is an exterior edge if and only if \( e \in E(G) \backslash ( \bigcup_{i=1}^k E(G_i)) \). The set of exterior edges will be denoted by $ E_\oplus $. A 2-edge-colored graph $G$ of order $2n$ is said to be an alternating-pancyclic graph, whenever for each $ l \in {2, . . ., n} $, there exists an alternating cycle of length $2l$ in $G$. The topics of pancyclism and vertex-pancyclism are deeply and widely studied by several authors. The existence of alternating cycles in 2-edge-colored graphs has been studied because of its many applications. In this paper, we give sufficient conditions for a graph $ G \in \oplus_{i=1}^k G_i $ to be an alternating-pancyclic graph.
Źródło:
Discussiones Mathematicae Graph Theory; 2021, 41, 3; 779-800
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Kernels and cycles subdivisions in arc-colored tournaments
Autorzy:
Delgado-Escalante, Pietra
Galeana-Sánchez, Hortensia
Powiązania:
https://bibliotekanauki.pl/articles/743117.pdf
Data publikacji:
2009
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
kernel
kernel by monochromatic paths
tournament
Opis:
Let D be a digraph. D is said to be an m-colored digraph if the arcs of D are colored with m colors. A path P in D is called monochromatic if all of its arcs are colored alike. Let D be an m-colored digraph. A set N ⊆ V(D) is said to be a kernel by monochromatic paths of D if it satisfies the following conditions: a) for every pair of different vertices u,v ∈ N there is no monochromatic directed path between them; and b) for every vertex x ∈ V(D)-N there is a vertex n ∈ N such that there is an xn-monochromatic directed path in D. In this paper we prove that if T is an arc-colored tournament which does not contain certain subdivisions of cycles then it possesses a kernel by monochromatic paths. These results generalize a well known sufficient condition for the existence of a kernel by monochromatic paths obtained by Shen Minggang in 1988 and another one obtained by Hahn et al. in 2004. Some open problems are proposed.
Źródło:
Discussiones Mathematicae Graph Theory; 2009, 29, 1; 101-117
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
On monochromatic paths and bicolored subdigraphs in arc-colored tournaments
Autorzy:
Delgado-Escalante, Pietra
Galeana-Sánchez, Hortensia
Powiązania:
https://bibliotekanauki.pl/articles/743637.pdf
Data publikacji:
2011
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
kernel
kernel by monochromatic paths
tournament
Opis:
Consider an arc-colored digraph. A set of vertices N is a kernel by monochromatic paths if all pairs of distinct vertices of N have no monochromatic directed path between them and if for every vertex v not in N there exists n ∈ N such that there is a monochromatic directed path from v to n. In this paper we prove different sufficient conditions which imply that an arc-colored tournament has a kernel by monochromatic paths. Our conditions concerns to some subdigraphs of T and its quasimonochromatic and bicolor coloration. We also prove that our conditions are not mutually implied and that they are not implied by those known previously. Besides some open problems are proposed.
Źródło:
Discussiones Mathematicae Graph Theory; 2011, 31, 4; 791-820
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:
A class of tight circulant tournaments
Autorzy:
Galeana-Sánchez, Hortensia
Neumann-Lara, Víctor
Powiązania:
https://bibliotekanauki.pl/articles/743727.pdf
Data publikacji:
2000
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
Circulant tournament
acyclic disconnection
vertex 3-colouring
3-chromatic triangle
tight tournament
Opis:
A tournament is said to be tight whenever every 3-colouring of its vertices using the 3 colours, leaves at least one cyclic triangle all whose vertices have different colours. In this paper, we extend the class of known tight circulant tournaments.
Źródło:
Discussiones Mathematicae Graph Theory; 2000, 20, 1; 109-128
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
A conjecture on cycle-pancyclism in tournaments
Autorzy:
Galeana-Sánchez, Hortensia
Rajsbaum, Sergio
Powiązania:
https://bibliotekanauki.pl/articles/744233.pdf
Data publikacji:
1998
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
Tournaments
pancyclism
cycle-pancyclism
Opis:
Let T be a hamiltonian tournament with n vertices and γ a hamiltonian cycle of T. In previous works we introduced and studied the concept of cycle-pancyclism to capture the following question: What is the maximum intersection with γ of a cycle of length k? More precisely, for a cycle Cₖ of length k in T we denote $I_γ (Cₖ) = |A(γ)∩A(Cₖ)|$, the number of arcs that γ and Cₖ have in common. Let $f(k,T,γ) = max{I_γ(Cₖ)|Cₖ ⊂ T}$ and f(n,k) = min{f(k,T,γ)|T is a hamiltonian tournament with n vertices, and γ a hamiltonian cycle of T}. In previous papers we gave a characterization of f(n,k). In particular, the characterization implies that f(n,k) ≥ k-4.
The purpose of this paper is to give some support to the following original conjecture: for any vertex v there exists a cycle of length k containing v with f(n,k) arcs in common with γ.
Źródło:
Discussiones Mathematicae Graph Theory; 1998, 18, 2; 243-251
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Cycle-pancyclism in bipartite tournaments I
Autorzy:
Galeana-Sánchez, Hortensia
Powiązania:
https://bibliotekanauki.pl/articles/744498.pdf
Data publikacji:
2004
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
bipartite tournament
pancyclism
Opis:
Let T be a hamiltonian bipartite tournament with n vertices, γ a hamiltonian directed cycle of T, and k an even number. In this paper, the following question is studied: What is the maximum intersection with γ of a directed cycle of length k? It is proved that for an even k in the range 4 ≤ k ≤ [(n+4)/2], there exists a directed cycle $C_{h(k)}$ of length h(k), h(k) ∈ {k,k-2} with $|A(C_{h(k)}) ∩ A(γ)| ≥ h(k)-3$ and the result is best possible.
In a forthcoming paper the case of directed cycles of length k, k even and k < [(n+4)/2] will be studied.
Źródło:
Discussiones Mathematicae Graph Theory; 2004, 24, 2; 277-290
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
  • odwiedzone
Tytuł:
Cycle-pancyclism in bipartite tournaments II
Autorzy:
Galeana-Sánchez, Hortensia
Powiązania:
https://bibliotekanauki.pl/articles/744263.pdf
Data publikacji:
2004
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
bipartite tournament
pancyclism
Opis:
Let T be a hamiltonian bipartite tournament with n vertices, γ a hamiltonian directed cycle of T, and k an even number. In this paper the following question is studied: What is the maximum intersection with γ of a directed cycle of length k contained in T[V(γ)]? It is proved that for an even k in the range (n+6)/2 ≤ k ≤ n-2, there exists a directed cycle $C_{h(k)}$ of length h(k), h(k) ∈ {k,k-2} with $|A(C_{h(k)}) ∩ A(γ)| ≥ h(k)-4$ and the result is best possible. In a previous paper a similar result for 4 ≤ k ≤ (n+4)/2 was proved.
Źródło:
Discussiones Mathematicae Graph Theory; 2004, 24, 3; 529-538
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Cyclically k-partite digraphs and k-kernels
Autorzy:
Galeana-Sánchez, Hortensia
Hernández-Cruz, César
Powiązania:
https://bibliotekanauki.pl/articles/743833.pdf
Data publikacji:
2011
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
digraph
kernel
(k,l)-kernel
k-kernel
cyclically k-partite
Opis:
Let D be a digraph, V(D) and A(D) will denote the sets of vertices and arcs of D, respectively.
A (k,l)-kernel N of D is a k-independent set of vertices (if u,v ∈ N then d(u,v) ≥ k) and l-absorbent (if u ∈ V(D)-N then there exists v ∈ N such that d(u,v) ≤ l). A k-kernel is a (k,k-1)-kernel. A digraph D is cyclically k-partite if there exists a partition ${V_i}_{i = 0}^{k-1}$ of V(D) such that every arc in D is a $V_i V_{i+1}-arc$ (mod k). We give a characterization for an unilateral digraph to be cyclically k-partite through the lengths of directed cycles and directed cycles with one obstruction, in addition we prove that such digraphs always have a k-kernel. A study of some structural properties of cyclically k-partite digraphs is made which bring interesting consequences, e.g., sufficient conditions for a digraph to have k-kernel; a generalization of the well known and important theorem that states if every cycle of a graph G has even length, then G is bipartite (cyclically 2-partite), we prove that if every cycle of a graph G has length ≡ 0 (mod k) then G is cyclically k-partite; and a generalization of another well known result about bipartite digraphs, a strong digraph D is bipartite if and only if every directed cycle has even length, we prove that an unilateral digraph D is bipartite if and only if every directed cycle with at most one obstruction has even length.
Źródło:
Discussiones Mathematicae Graph Theory; 2011, 31, 1; 63-78
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Directed hypergraphs: a tool for researching digraphs and hypergraphs
Autorzy:
Galeana-Sánchez, Hortensia
Manrique, Martín
Powiązania:
https://bibliotekanauki.pl/articles/743191.pdf
Data publikacji:
2009
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
hypergraph
strongly independent set
transversal set
kernel
Opis:
In this paper we introduce the concept of directed hypergraph. It is a generalisation of the concept of digraph and is closely related with hypergraphs. The basic idea is to take a hypergraph, partition its edges non-trivially (when possible), and give a total order to such partitions. The elements of these partitions are called levels. In order to preserve the structure of the underlying hypergraph, we ask that only vertices which belong to exactly the same edges may be in the same level of any edge they belong to. Some little adjustments are needed to avoid directed walks within a single edge of the underlying hypergraph, and to deal with isolated vertices.
The concepts of independent set, absorbent set, and transversal set are inherited directly from digraphs.
As a consequence of our results on this topic, we have found both a class of kernel-perfect digraphs with odd cycles and a class of hypergraphs which have a strongly independent transversal set.
Źródło:
Discussiones Mathematicae Graph Theory; 2009, 29, 2; 313-335
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Independent transversals of longest paths in locally semicomplete and locally transitive digraphs
Autorzy:
Galeana-Sánchez, Hortensia
Gómez, Ricardo
Montellano-Ballesteros, Juan
Powiązania:
https://bibliotekanauki.pl/articles/744431.pdf
Data publikacji:
2009
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
independent set
longest path
locally semicomplete
locally transitive
Opis:
We present several results concerning the Laborde-Payan-Xuang conjecture stating that in every digraph there exists an independent set of vertices intersecting every longest path. The digraphs we consider are defined in terms of local semicompleteness and local transitivity. We also look at oriented graphs for which the length of a longest path does not exceed 4.
Źródło:
Discussiones Mathematicae Graph Theory; 2009, 29, 3; 469-480
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
k-Kernels and some operations in digraphs
Autorzy:
Galeana-Sanchez, Hortensia
Pastrana, Laura
Powiązania:
https://bibliotekanauki.pl/articles/743109.pdf
Data publikacji:
2009
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
k-kernel
k-subdivision digraph
k-middle digraph and k-total digraph
Opis:
Let D be a digraph. V(D) denotes the set of vertices of D; a set N ⊆ V(D) is said to be a k-kernel of D if it satisfies the following two conditions: for every pair of different vertices u,v ∈ N it holds that every directed path between them has length at least k and for every vertex x ∈ V(D)-N there is a vertex y ∈ N such that there is an xy-directed path of length at most k-1. In this paper, we consider some operations on digraphs and prove the existence of k-kernels in digraphs formed by these operations from another digraphs.
Źródło:
Discussiones Mathematicae Graph Theory; 2009, 29, 1; 39-49
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
k-kernels in generalizations of transitive digraphs
Autorzy:
Galeana-Sánchez, Hortensia
Hernández-Cruz, César
Powiązania:
https://bibliotekanauki.pl/articles/743887.pdf
Data publikacji:
2011
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
digraph
kernel
(k,l)-kernel
k-kernel
transitive digraph
quasi-transitive digraph
right-pretransitive digraph
left-pretransitive digraph
pretransitive digraph
Opis:
Let D be a digraph, V(D) and A(D) will denote the sets of vertices and arcs of D, respectively.
A (k,l)-kernel N of D is a k-independent set of vertices (if u,v ∈ N, u ≠ v, then d(u,v), d(v,u) ≥ k) and l-absorbent (if u ∈ V(D)-N then there exists v ∈ N such that d(u,v) ≤ l). A k-kernel is a (k,k-1)-kernel. Quasi-transitive, right-pretransitive and left-pretransitive digraphs are generalizations of transitive digraphs. In this paper the following results are proved: Let D be a right-(left-) pretransitive strong digraph such that every directed triangle of D is symmetrical, then D has a k-kernel for every integer k ≥ 3; the result is also valid for non-strong digraphs in the right-pretransitive case. We also give a proof of the fact that every quasi-transitive digraph has a (k,l)-kernel for every integers k > l ≥ 3 or k = 3 and l = 2.
Źródło:
Discussiones Mathematicae Graph Theory; 2011, 31, 2; 293-312
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Kernels by Monochromatic Paths and Color-Perfect Digraphs
Autorzy:
Galeana-Śanchez, Hortensia
Sánchez-López, Rocío
Powiązania:
https://bibliotekanauki.pl/articles/31340961.pdf
Data publikacji:
2016-05-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
kernel
kernel perfect digraph
kernel by monochromatic paths
color-class digraph
quasi color-perfect digraph
color-perfect digraph
Opis:
For a digraph D, V (D) and A(D) will denote the sets of vertices and arcs of D respectively. In an arc-colored digraph, a subset K of V(D) is said to be kernel by monochromatic paths (mp-kernel) if (1) for any two different vertices x, y in N there is no monochromatic directed path between them (N is mp-independent) and (2) for each vertex u in V (D) \ N there exists v ∈ N such that there is a monochromatic directed path from u to v in D (N is mp-absorbent). If every arc in D has a different color, then a kernel by monochromatic paths is said to be a kernel. Two associated digraphs to an arc-colored digraph are the closure and the color-class digraph C(D). In this paper we will approach an mp-kernel via the closure of induced subdigraphs of D which have the property of having few colors in their arcs with respect to D. We will introduce the concept of color-perfect digraph and we are going to prove that if D is an arc-colored digraph such that D is a quasi color-perfect digraph and C(D) is not strong, then D has an mp-kernel. Previous interesting results are generalized, as for example Richardson′s Theorem.
Źródło:
Discussiones Mathematicae Graph Theory; 2016, 36, 2; 309-321
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
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