$H$-kernels in $H$-colored digraphs without $(ξ_{1}, ξ, ξ_{2})$-$H$-subdivisions of $\overrightarrow{C_{3}}$

Abstract

Let H be a digraph possibly with loops and D a digraph without loops with a coloring of its arcs c : A(D) → V (H) (D is said to be an H-colored digraph). A directed path W in D is said to be an H-path if and only if the consecutive colors encountered on W form a directed walk in H. A subset N of vertices of D is said to be an H-kernel if (1) for every pair of different vertices in N there is no H-path between them and (2) for every vertex u in V(D)\N there exists an H-path in D from u to N . Under this definition an H-kernel is a kernel whenever A(H) = ∅. The color-class digraph CC (D) of D is the digraph whose vertices are the colors represented in the arcs of D and (i,j) ∈ A(CC (D)) if and only if there exist two arcs, namely (u,v) and (v,w) in D, such that (u,v) has color i and (v,w) has color j. Since not every H-colored digraph has an H-kernel and V (CC (D)) = V (H), the natural question is: what structural properties of CC (D), with respect to the H-coloring, imply that D has an H-kernel? In this paper we investigate the problem of the existence of an H-kernel by means of a partition ξ of V (H) and a partition {ξ1, ξ2} of ξ. We establish conditions on the directed cycles and the directed paths of the digraph D, with respect to the partition {ξ1, ξ2}. In particular we pay attention to some subestructures produced by the partitions ξ and {ξ1, ξ2}, namely (ξ1, ξ, ξ2)-H-subdivisions of − → C3 and (ξ1, ξ, ξ2)-H-subdivisions of − → P3. We give some examples which show that each hypothesis in the main result is tight.