Elsevier

Discrete Mathematics

Volume 308, Issue 20, 28 October 2008, Pages 4599-4607
Discrete Mathematics

Kernels in edge-coloured orientations of nearly complete graphs

https://doi.org/10.1016/j.disc.2007.08.079Get rights and content
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Abstract

We call the digraph D an orientation of a graph G if D is obtained from G by the orientation of each edge of G in exactly one of the two possible directions. The digraph D is an m-coloured digraph if the arcs of D are coloured with m-colours.

Let D be an m-coloured digraph. A directed path (or a directed cycle) is called monochromatic if all of its arcs are coloured alike.

A set NV(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,vN there is no monochromatic directed path between them and (ii) for every vertex xV(D)-N there is a vertex yN such that there is an xy-monochromatic directed path.

In this paper we obtain sufficient conditions for an m-coloured orientation of a graph obtained from Kn by deletion of the arcs of K1,r (0rn-1) to have a kernel by monochromatic.

MSC

05C20

Keywords

Kernel
Kernel by monochromatic paths
Orientation

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