New classes of panchromatic digraphs

A digraph D=(V,A) with a k-colouring of its arcs ς:A→[k] is said to have a ς-kernel if there exists a subset K of V such that there are no monochromatic uv-paths for any two vertices u,v∈K, but for every w∈V−K, there exists a vertex v∈K such that there is a monochromatic wv-path in D. The panchromatic number, π(D), of D is the greatest integer k for which D has a ς-kernel for every possible k-colouring of its arcs. D is said to be a panchromatic digraph if, for every k≤|A| and every k-colouring ς:A→[k], D has a ς-kernel. In this paper we study the panchromaticity of cycles. In particular, we show that even cycles are panchromatic and that π(C)=2 when C is an odd cycle. We also set sufficient conditions, in terms of its induced subdigraphs, for a digraph D to be panchromatic, and we show through counterexamples that these results cannot be improved