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Cycle-pancyclism in bipartite tournaments I

100%

Galeana-Sánchez H.

Discussiones Mathematicae Graph Theory

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2004

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tom 24

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nr 2

277-290

EN

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.

2

Kernels by monochromatic paths and the color-class digraph

100%

Galeana-Sánchez H.

Discussiones Mathematicae Graph Theory

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2011

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tom 31

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nr 2

273-281

EN

An m-colored digraph is a digraph whose arcs are colored with m colors. A directed path is monochromatic when its arcs are colored alike. A set S ⊆ V(D) is a kernel by monochromatic paths whenever the two following conditions hold: 1. For any x,y ∈ S, x ≠ y, there is no monochromatic directed path between them. 2. For each z ∈ (V(D)-S) there exists a zS-monochromatic directed path. In this paper it is introduced the concept of color-class digraph to prove that if D is an m-colored strongly connected finite digraph such that: (i) Every closed directed walk has an even number of color changes, (ii) Every directed walk starting and ending with the same color has an even number of color changes, then D has a kernel by monochromatic paths. This result generalizes a classical result by Sands, Sauer and Woodrow which asserts that any 2-colored digraph has a kernel by monochromatic paths, in case that the digraph D be a strongly connected digraph.

3

Cycle-pancyclism in bipartite tournaments II

100%

Galeana-Sánchez H.

Discussiones Mathematicae Graph Theory

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2004

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tom 24

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nr 3

529-538

EN

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.

4

Some sufficient conditions on odd directed cycles of bounded length for the existence of a kernel

100%

Galeana-Sánchez H.

Discussiones Mathematicae Graph Theory

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2004

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tom 24

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nr 2

171-182

EN

A kernel N of a digraph D is an independent set of vertices of D such that for every w ∈ V(D)-N there exists an arc from w to N. If every induced subdigraph of D has a kernel, D is said to be a kernel-perfect digraph. In this paper I investigate some sufficient conditions for a digraph to have a kernel by asking for the existence of certain diagonals or symmetrical arcs in each odd directed cycle whose length is at most 2α(D)+1, where α(D) is the maximum cardinality of an independent vertex set of D. Previous results are generalized.

5

Kernels in the closure of coloured digraphs

64%

Galeana-Sánchez H., García-Ruvalcaba J.

Discussiones Mathematicae Graph Theory

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2000

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tom 20

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nr 2

243-254

EN

Let D be a digraph with V(D) and A(D) the sets of vertices and arcs of D, respectively. A kernel of D is a set I ⊂ V(D) such that no arc of D joins two vertices of I and for each x ∈ V(D)∖I there is a vertex y ∈ I such that (x,y) ∈ A(D). A digraph is kernel-perfect if every non-empty induced subdigraph of D has a kernel. If D is edge coloured, we define the closure ξ(D) of D the multidigraph with V(ξ(D)) = V(D) and $A(ξ(D)) = ⋃_i{(u,v)$ with colour i there exists a monochromatic path of colour i from the vertex u to the vertex v contained in D}. Let T₃ and C₃ denote the transitive tournament of order 3 and the 3-cycle, respectively, both of whose arcs are coloured with 3 different colours. In this paper, we survey sufficient conditions for the existence of kernels in the closure of edge coloured digraphs, also we prove that if D is obtained from an edge coloured tournament by deleting one arc and D does not contain T₃ or C₃, then ξ(D) is a kernel-perfect digraph.

6

A class of tight circulant tournaments

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Galeana-Sánchez H., Neumann-Lara V.

Discussiones Mathematicae Graph Theory

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2000

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tom 20

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nr 1

109-128

EN

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.

7

On graphs all of whose {C₃,T₃}-free arc colorations are kernel-perfect

64%

Galeana-Sánchez H., García-Ruvalcaba J.

Discussiones Mathematicae Graph Theory

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2001

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tom 21

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nr 1

77-93

EN

A digraph D is called a kernel-perfect digraph or KP-digraph when every induced subdigraph of D has a kernel. We call the digraph D an m-coloured digraph if the arcs of D are coloured with m distinct colours. A path P is monochromatic in D if all of its arcs are coloured alike in D. The closure of D, denoted by ζ(D), is the m-coloured digraph defined as follows: V( ζ(D)) = V(D), and A( ζ(D)) = ∪_{i} {(u,v) with colour i: there exists a monochromatic path of colour i from the vertex u to the vertex v contained in D}. We will denoted by T₃ and C₃, the transitive tournament of order 3 and the 3-directed-cycle respectively; both of whose arcs are coloured with three different colours. Let G be a simple graph. By an m-orientation-coloration of G we mean an m-coloured digraph which is an asymmetric orientation of G. By the class E we mean the set of all the simple graphs G that for any m-orientation-coloration D without C₃ or T₃, we have that ζ(D) is a KP-digraph. In this paper we prove that if G is a hamiltonian graph of class E, then its complement has at most one nontrivial component, and this component is K₃ or a star.

8

New classes of critical kernel-imperfect digraphs

64%

Galeana-Sánchez H., Neumann-Lara V.

Discussiones Mathematicae Graph Theory

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1998

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tom 18

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nr 1

85-89

EN

A kernel of a digraph D is a subset N ⊆ V(D) which is both independent and absorbing. When every induced subdigraph of D has a kernel, the digraph D is said to be kernel-perfect. We say that D is a critical kernel-imperfect digraph if D does not have a kernel but every proper induced subdigraph of D does have at least one. Although many classes of critical kernel-imperfect-digraphs have been constructed, all of them are digraphs such that the block-cutpoint tree of its asymmetrical part is a path. The aim of the paper is to construct critical kernel-imperfect digraphs of a special structure, a general method is developed which permits to build critical kernel-imperfect-digraphs whose asymmetrical part has a prescribed block-cutpoint tree. Specially, any directed cactus (an asymmetrical digraph all of whose blocks are directed cycles) whose blocks are directed cycles of length at least 5 is the asymmetrical part of some critical kernel-imperfect-digraph.

9

A conjecture on cycle-pancyclism in tournaments

64%

Galeana-Sánchez H., Rajsbaum S.

Discussiones Mathematicae Graph Theory

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1998

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tom 18

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nr 2

243-251

EN

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.

10

(k,l)-kernels, (k,l)-semikernels, k-Grundy functions and duality for state splittings

64%

Galeana-Sánchez H., Gómez R.

Discussiones Mathematicae Graph Theory

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2007

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tom 27

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nr 2

359-371

EN

Line digraphs can be obtained by sequences of state splittings, a particular kind of operation widely used in symbolic dynamics [12]. Properties of line digraphs inherited from the source have been studied, for instance in [7] Harminc showed that the cardinalities of the sets of kernels and solutions (kernel's dual definition) of a digraph and its line digraph coincide. We extend this for (k,l)-kernels in the context of state splittings and also look at (k,l)-semikernels, k-Grundy functions and their duals.

11

Directed hypergraphs: a tool for researching digraphs and hypergraphs

64%

Galeana-Sánchez H., Manrique M.

Discussiones Mathematicae Graph Theory

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2009

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tom 29

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nr 2

313-335

EN

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.

12

Kernels and cycles' subdivisions in arc-colored tournaments

64%

Delgado-Escalante P., Galeana-Sánchez H.

Discussiones Mathematicae Graph Theory

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2009

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tom 29

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nr 1

101-117

EN

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.

13

Monochromatic kernel-perfectness of special classes of digraphs

64%

Galeana-Sánchez H., Ramírez L.

Discussiones Mathematicae Graph Theory

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2007

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tom 27

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nr 3

389-400

EN

In this paper, we introduce the concept of monochromatic kernel-perfect digraph, and we prove the following two results: (1) If D is a digraph without monochromatic directed cycles, then D and each $α_v,v ∈ V(D)$ are monochromatic kernel-perfect digraphs if and only if the composition over D of $(α_v)_{v ∈ V(D)}$ is a monochromatic kernel-perfect digraph. (2) D is a monochromatic kernel-perfect digraph if and only if for any B ⊆ V(D), the duplication of D over B, $D^B$, is a monochromatic kernel-perfect digraph.

14

k-kernels in generalizations of transitive digraphs

64%

Galeana-Sánchez H., Hernández-Cruz C.

Discussiones Mathematicae Graph Theory

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2011

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tom 31

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nr 2

293-312

EN

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.

15

On the heterochromatic number of circulant digraphs

64%

Galeana-Sánchez H., Neumann-Lara V.

Discussiones Mathematicae Graph Theory

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2004

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tom 24

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nr 1

73-79

EN

The heterochromatic number hc(D) of a digraph D, is the minimum integer k such that for every partition of V(D) into k classes, there is a cyclic triangle whose three vertices belong to different classes. For any two integers s and n with 1 ≤ s ≤ n, let $D_{n,s}$ be the oriented graph such that $V(D_{n,s})$ is the set of integers mod 2n+1 and $A(D_{n,s}) = {(i,j) : j-i ∈ {1,2,...,n}∖{s}}.. In this paper we prove that $hc(D_{n,s}) ≤ 5$ for n ≥ 7. The bound is tight since equality holds when s ∈ {n,[(2n+1)/3]}.

16

Cyclically k-partite digraphs and k-kernels

64%

Galeana-Sánchez H., Hernández-Cruz C.

Discussiones Mathematicae Graph Theory

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2011

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tom 31

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nr 1

63-78

EN

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.

17

On monochromatic paths and bicolored subdigraphs in arc-colored tournaments

64%

Delgado-Escalante P., Galeana-Sánchez H.

Discussiones Mathematicae Graph Theory

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2011

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tom 31

|

nr 4

791-820

EN

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.

18

γ-Cycles And Transitivity By Monochromatic Paths In Arc-Coloured Digraphs

52%

Casas-Bautista E., Galeana-Sánchez H., Rojas-Monroy R.

Discussiones Mathematicae Graph Theory

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2013

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tom 33

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nr 3

493-507

EN

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.

19

γ-Cycles In Arc-Colored Digraphs

52%

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

Discussiones Mathematicae Graph Theory

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2016

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tom 36

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nr 1

103-116

EN

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.

20

Independent transversals of longest paths in locally semicomplete and locally transitive digraphs

52%

Galeana-Sánchez H., Gómez R., Montellano-Ballesteros J.

Discussiones Mathematicae Graph Theory

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2009

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tom 29

|

nr 3

469-480

EN

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.

Ograniczanie wyników

Cycle-pancyclism in bipartite tournaments I

Kernels by monochromatic paths and the color-class digraph

Cycle-pancyclism in bipartite tournaments II

Some sufficient conditions on odd directed cycles of bounded length for the existence of a kernel

Kernels in the closure of coloured digraphs

A class of tight circulant tournaments

On graphs all of whose {C₃,T₃}-free arc colorations are kernel-perfect

New classes of critical kernel-imperfect digraphs

A conjecture on cycle-pancyclism in tournaments

(k,l)-kernels, (k,l)-semikernels, k-Grundy functions and duality for state splittings

Directed hypergraphs: a tool for researching digraphs and hypergraphs

Kernels and cycles' subdivisions in arc-colored tournaments

Monochromatic kernel-perfectness of special classes of digraphs

k-kernels in generalizations of transitive digraphs

On the heterochromatic number of circulant digraphs

Cyclically k-partite digraphs and k-kernels

On monochromatic paths and bicolored subdigraphs in arc-colored tournaments

γ-Cycles And Transitivity By Monochromatic Paths In Arc-Coloured Digraphs

γ-Cycles In Arc-Colored Digraphs

Independent transversals of longest paths in locally semicomplete and locally transitive digraphs