On kernels by rainbow paths in arc-coloured digraphs
Abstract
In 2018, Bai, Fujita and Zhang (\emph{Discrete Math.} 2018, 341(6): 1523-1533) introduced the concept of a kernel by rainbow paths (for short, RP-kernel) of an arc-coloured digraph , which is a subset of vertices of such that () there exists no rainbow path for any pair of distinct vertices of , and () every vertex outside can reach by a rainbow path in . They showed that it is NP-hard to recognize wether an arc-coloured digraph has a RP-kernel and it is NP-complete to decided wether an arc-coloured tournament has a RP-kernel. In this paper, we give the sufficient conditions for the existence of a RP-kernel in arc-coloured unicyclic digraphs, semicomplete digraphs, quasi-transitive digraphs and bipartite tournaments, and prove that these arc-coloured digraphs have RP-kernels if certain "short" cycles and certain "small" induced subdigraphs are rainbow.
Cite
@article{arxiv.1807.08286,
title = {On kernels by rainbow paths in arc-coloured digraphs},
author = {Ruijuan Li and Yanqin Cao},
journal= {arXiv preprint arXiv:1807.08286},
year = {2018}
}