On kernels by rainbow paths in arc-coloured digraphs

TL;DR

This paper investigates conditions under which arc-coloured digraphs possess a rainbow kernel, providing sufficient criteria for various classes of digraphs and extending understanding of rainbow path structures.

Contribution

It introduces new sufficient conditions for the existence of rainbow kernels in specific classes of arc-coloured digraphs, expanding prior NP-hardness results.

Findings

RP-kernels exist in certain unicyclic digraphs under cycle conditions

RP-kernels are guaranteed in semicomplete digraphs with specific subdigraphs

Conditions involving small cycles ensure rainbow kernels in bipartite tournaments

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 $D$, which is a subset $S$ of vertices of $D$ such that ($a$) there exists no rainbow path for any pair of distinct vertices of $S$, and ($b$) every vertex outside $S$ can reach $S$ by a rainbow path in $D$. 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.