# Kernels by properly colored paths in arc-colored digraphs

## TL;DR

This paper investigates the existence of kernels by properly colored paths in arc-colored digraphs, proposing a conjecture and verifying it for specific classes like unicyclic digraphs, semi-complete digraphs, and bipartite tournaments.

## Contribution

It introduces a conjecture on kernels by properly colored paths in all such digraphs and confirms it for several important classes, providing weaker conditions for some cases.

## Key findings

- Conjecture that all cycles properly colored imply such kernels.
- Verified the conjecture for unicyclic digraphs.
- Confirmed the conjecture for semi-complete digraphs and bipartite tournaments.

## Abstract

A {\em kernel by properly colored paths} of an arc-colored digraph $D$ is a set $S$ of vertices of $D$ such that (i) no two vertices of $S$ are connected by a properly colored directed path in $D$, and (ii) every vertex outside $S$ can reach $S$ by a properly colored directed path in $D$. In this paper, we conjecture that every arc-colored digraph with all cycles properly colored has such a kernel and verify the conjecture for unicyclic digraphs, semi-complete digraphs and bipartite tournaments, respectively. Moreover, weaker conditions for the latter two classes of digraphs are given.