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k-Kernels and some operations in digraphs

100%

Galeana-Sanchez H., Pastrana L.

Discussiones Mathematicae Graph Theory

2009

tom 29

nr 1

39-49

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.

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

100%

Galeana-Sanchez H., Rojas-Monroy R.

Discussiones Mathematicae Graph Theory

2008

tom 28

nr 2

285-306

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 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 xy-monochromatic directed path. In this paper it is proved that if D is an m-coloured bipartite tournament such that: every directed cycle of length 4 is quasi-monochromatic, every directed cycle of length 6 is monochromatic, and D has no induced particular 6-element bipartite tournament T̃₆, then D has a kernel by monochromatic paths.

Ograniczanie wyników

2 Discussiones Mathematicae Graph Theory

2 Galeana-Sanchez H.

1 Pastrana L.

1 Rojas-Monroy R.

1 2009

1 2008

Wyniki wyszukiwania

k-Kernels and some operations in digraphs

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