On Panchromatic Patterns

TL;DR

This paper characterizes panchromatic patterns in coloured digraphs, solving a problem from 2007 by identifying which patterns guarantee kernels by H-walks in all coloured digraphs.

Contribution

It provides a complete characterization of panchromatic patterns, advancing understanding of kernels in coloured digraphs and resolving a longstanding open problem.

Findings

Identified all panchromatic patterns that guarantee kernels by H-walks.

Solved the 2007 problem posed by Arpin and Linek.

Established conditions for the existence of kernels in coloured digraphs.

Abstract

Given D and H two digraphs, D is H-coloured iff the arcs of D are coloured with the vertices of H. After defining what do we mean by an H-walk in the coloured D, we characterise those H, which we call panchromatic patterns, for which all D and all H-colourings of D admit a kernel by H-walks. This solves a problem of Arpin and Linek from 2007.