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Kernels in the closure of coloured digraphs

100%
Galeana-Sánchez H., García-Ruvalcaba J.
Discussiones Mathematicae Graph Theory
|
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.
2
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The Dichromatic Number of Infinite Families of Circulant Tournaments

100%
Javier N., Llano B.
Discussiones Mathematicae Graph Theory
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2017
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tom 37
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nr 1
221-238
EN
The dichromatic number dc(D) of a digraph D is defined to be the minimum number of colors such that the vertices of D can be colored in such a way that every chromatic class induces an acyclic subdigraph in D. The cyclic circulant tournament is denoted by [...] T=C→2n+1(1,2,…,n) $T = \overrightarrow C _{2n + 1} (1,2, \ldots ,n)$ , where V (T) = ℤ2n+1 and for every jump j ∈ {1, 2, . . . , n} there exist the arcs (a, a + j) for every a ∈ ℤ2n+1. Consider the circulant tournament [...] C→2n+1〈k〉 $\overrightarrow C _{2n + 1} \left\langle k \right\rangle $ obtained from the cyclic tournament by reversing one of its jumps, that is, [...] C→2n+1 〈k〉 $\overrightarrow C _{2n + 1} \left\langle k \right\rangle $ has the same arc set as [...] C→2n+1(1,2,…,n) $\overrightarrow C _{2n + 1} (1,2, \ldots ,n)$ except for j = k in which case, the arcs are (a, a − k) for every a ∈ ℤ2n+1. In this paper, we prove that [...] dc(C→2n+1 〈k〉)∈{2,3,4} $dc ( {\overrightarrow C _{2n + 1} \left\langle k \right\rangle } ) \in \{ 2,3,4\}$ for every k ∈ {1, 2, . . . , n}. Moreover, we classify which circulant tournaments [...] C→2n+1 〈k〉 $\overrightarrow C _{2n + 1} \left\langle k \right\rangle$ are vertex-critical r-dichromatic for every k ∈ {1, 2, . . . , n} and r ∈ {2, 3, 4}. Some previous results by Neumann-Lara are generalized.
3
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An equational basis in four variables for the three-element tournament

100%
Grätzer G., Kisielewicz A., Wolk B.
Colloquium Mathematicum
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1992
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tom 63
|
nr 1
41-44
4
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Kernels and cycles' subdivisions in arc-colored tournaments

100%
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.
5
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On a conjecture of quintas and arc-traceability in upset tournaments

100%
Busch A., Jacobson M., Reid K.
Discussiones Mathematicae Graph Theory
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2005
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tom 25
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nr 3
225-239
EN
A digraph D = (V,A) is arc-traceable if for each arc xy in A, xy lies on a directed path containing all the vertices of V, i.e., hamiltonian path. We prove a conjecture of Quintas [7]: if D is arc-traceable, then the condensation of D is a directed path. We show that the converse of this conjecture is false by providing an example of an upset tournament which is not arc-traceable. We then give a characterization for upset tournaments in terms of their score sequences, characterize which arcs of an upset tournament lie on a hamiltonian path, and deduce a characterization of arc-traceable upset tournaments.
6
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On monochromatic paths and bicolored subdigraphs in arc-colored tournaments

100%
Delgado-Escalante P., Galeana-Sánchez H.
Discussiones Mathematicae Graph Theory
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2011
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tom 31
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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.
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