On panchromatic patterns

Abstract

Since the classic book of Berge (1985) it is well known that every digraph contains a kernel by paths. This was generalised by Sands et al. (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).