Displaying similar documents to “Kernels and cycles' subdivisions in arc-colored tournaments”

On monochromatic paths and bicolored subdigraphs in arc-colored tournaments

Pietra Delgado-Escalante, Hortensia Galeana-Sánchez (2011)

Discussiones Mathematicae Graph Theory

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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...

γ-Cycles In Arc-Colored Digraphs

Hortensia Galeana-Sánchez, Guadalupe Gaytán-Gómez, Rocío Rojas-Monroy (2016)

Discussiones Mathematicae Graph Theory

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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...

Monochromatic paths and monochromatic sets of arcs in bipartite tournaments

Hortensia Galeana-Sánchez, R. Rojas-Monroy, B. Zavala (2009)

Discussiones Mathematicae Graph Theory

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We call the digraph D an m-coloured digraph if the arcs of D are coloured with m colours and all of them are used. A directed path is called monochromatic if all of its arcs are coloured alike. A set N of vertices of D is called a kernel by monochromatic paths if for every pair of vertices there is no monochromatic path between them and for every vertex v in V(D)∖N there is a monochromatic path from v to some vertex in N. We denote by A⁺(u) the set of arcs of D that have u as...

Kernels in monochromatic path digraphs

Hortensia Galeana-Sánchez, Laura Pastrana Ramírez, Hugo Alberto Rincón Mejía (2005)

Discussiones Mathematicae Graph Theory

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We call the digraph D an m-coloured digraph if its arcs are coloured with m colours. A directed path (or a directed cycle) is called monochromatic if all of its arcs are coloured alike. Let D be an m-coloured digraph. 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 each vertex x ∈...

Monochromatic paths and monochromatic sets of arcs in 3-quasitransitive digraphs

Hortensia Galeana-Sánchez, R. Rojas-Monroy, B. Zavala (2009)

Discussiones Mathematicae Graph Theory

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We call the digraph D an m-coloured digraph if the arcs of D are coloured with m colours. A directed path is called monochromatic if all of its arcs are coloured alike. A set N of vertices of D is called a kernel by monochromatic paths if for every pair of vertices of N there is no monochromatic path between them and for every vertex v ∉ N there is a monochromatic path from v to N. We denote by A⁺(u) the set of arcs of D that have u as the initial vertex. We prove that if D is an m-coloured...

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

Enrique Casas-Bautista, Hortensia Galeana-Sánchez, Rocío Rojas-Monroy (2013)

Discussiones Mathematicae Graph Theory

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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...

Kernels in edge coloured line digraph

H. Galeana-Sánchez, L. Pastrana Ramírez (1998)

Discussiones Mathematicae Graph Theory

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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 set N ⊆ V(D) is said to be a kernel by monochromatic paths if it satisfies the two following 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...

Kernels by monochromatic paths and the color-class digraph

Hortensia Galeana-Sánchez (2011)

Discussiones Mathematicae Graph Theory

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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...

Monochromatic paths and quasi-monochromatic cycles in edge-coloured bipartite tournaments

Hortensia Galeana-Sanchez, Rocío Rojas-Monroy (2008)

Discussiones Mathematicae Graph Theory

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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...

Kernels by Monochromatic Paths and Color-Perfect Digraphs

Hortensia Galeana-Śanchez, Rocío Sánchez-López (2016)

Discussiones Mathematicae Graph Theory

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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...

On kernels by monochromatic paths in the corona of digraphs

Iwona Włoch (2008)

Open Mathematics

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In this paper we derive necessary and sufficient conditions for the existence of kernels by monochromatic paths in the corona of digraphs. Using these results, we are able to prove the main result of this paper which provides necessary and sufficient conditions for the corona of digraphs to be monochromatic kernel-perfect. Moreover we calculate the total numbers of kernels by monochromatic paths, independent by monochromatic paths sets and dominating by monochromatic paths sets in this...

Monochromatic cycles and monochromatic paths in arc-colored digraphs

Hortensia Galeana-Sánchez, Guadalupe Gaytán-Gómez, Rocío Rojas-Monroy (2011)

Discussiones Mathematicae Graph Theory

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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...

Monochromatic paths and monochromatic sets of arcs in quasi-transitive digraphs

Hortensia Galeana-Sánchez, R. Rojas-Monroy, B. Zavala (2010)

Discussiones Mathematicae Graph Theory

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Let D be a digraph, V(D) and A(D) will denote the sets of vertices and arcs of D, respectively. We call the digraph D an m-coloured digraph if each arc of D is coloured by an element of {1,2,...,m} where m ≥ 1. A directed path is called monochromatic if all of its arcs are coloured alike. A set N of vertices of D is called a kernel by monochromatic paths if there is no monochromatic path between two vertices of N and if for every vertex v not in N there is a monochromatic path from v...

Exponents of two-colored digraphs

Yan Ling Shao, Yubin Gao (2009)

Czechoslovak Mathematical Journal

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We consider the primitive two-colored digraphs whose uncolored digraph has n + s vertices and consists of one n -cycle and one ( n - 3 ) -cycle. We give bounds on the exponents and characterizations of extremal two-colored digraphs.

Rainbow Connectivity of Cacti and of Some Infinite Digraphs

Jesús Alva-Samos, Juan José Montellano-Ballesteros (2017)

Discussiones Mathematicae Graph Theory

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An arc-coloured digraph D = (V,A) is said to be rainbow connected if for every pair {u, v} ⊆ V there is a directed uv-path all whose arcs have different colours and a directed vu-path all whose arcs have different colours. The minimum number of colours required to make the digraph D rainbow connected is called the rainbow connection number of D, denoted rc⃗ (D). A cactus is a digraph where each arc belongs to exactly one directed cycle. In this paper we give sharp upper and lower bounds...

k-Kernels and some operations in digraphs

Hortensia Galeana-Sanchez, Laura Pastrana (2009)

Discussiones Mathematicae Graph Theory

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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...

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

Hortensia Galeana-Sánchez, José de Jesús García-Ruvalcaba (2001)

Discussiones Mathematicae Graph Theory

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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...