Formula omitted]-panchromatic digraphs

Abstract

Let H and D be two digraphs; D without loops or multiple arcs. An H−coloring of D is a function ρ:A(D)→V(H). We say that D is an (H,ρ)−colored digraph. For an arc (x,y) of D, we say that ρ(x,y) is the color of (x,y) over the H−coloring ρ.A directed path (x1,…,xn) in D is an (H,ρ)−path if (ρ(x1,x2),…,ρ(xn−1,xn)) is a directed walk in H. An (H,ρ)−kernel in an (H,ρ)−colored digraph is a subset of vertices of D, say S, such that for every pair of different vertices in S there is no (H,ρ)−path between them and every vertex outside S can reach S by an (H,ρ)−path. A digraph D is an ℋ-panchromatic digraph if D has an (H,ρ)−kernel for every digraph H and every H−coloring ρ of D. In this paper we show that ℋ-panchromatic digraphs cannot be characterized by means of certain forbidden subdigraphs. Also we will show ℋ-panchromaticity of some classes of digraphs and we show that ℋ-panchromaticity can be hereditary in some operations of digraphs.