-panchromatic digraphs. Issue 1 (2nd January 2020) - Galeana-Sánchez, Hortensia; Tecpa-Galván, Miguel
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 ( x 1, …, x n ) in D is an ( H, ρ ) − path if ( ρ ( x 1, x 2 ), …, ρ ( x n − 1, x n ) ) 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.