On panchromatic patterns

Since the classic book of Berge (1985) it is well known that every digraph contains a kernel by paths. This was generalised by Sands etźal. (1982) who proved that every edge two-coloured digraph has a kernel by monochromatic paths. More generally, given D and H two digraphs, D is H -coloured iff the arcs of D are coloured with the vertices of H . Furthermore, an H -walk in D is a sequence of arcs forming a walk in D whose colours are a walk in H . With this notion of H -walks, we can define H -independence, which is the absence of such a walk pairwise, and H -absorbance, which is the existence of such a walk towards the absorbent set. Thus, an H -kernel is a subset of vertices which is both H -independent and H -absorbent. The aim of this paper is to characterise those H , which we call panchromatic patterns, for which all D and all H -colourings of D admits an H -kernel. This solves a problem of Arpin and Linek from 2007 (Arpin and Linek, 2007).