$H$-kernels in $H$-colored digraphs without $(\xi_{1}, \xi, \xi_{2})$-$H$-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 $c:A(D) \rightarrow V(H)$ ($D$ 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($D$)$\setminus$$N$ there exists an $H$-path in $D$ from $u$ to $N$. Under this definition an $H$-kernel is a kernel whenever $A(H)=\emptyset$. The color-class digraph $\mathscr{C}_C$($D$) of $D$ is the digraph whose vertices are the colors represented in the arcs of $D$ and ($i$,$j$) $\in$ $A$($\mathscr{C}_C$($D$)) 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 $V(\mathscr{C}_C(D))= V(H)$, the natural question is: what structural properties of $\mathscr{C}_C(D)$, 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 $\xi$ of $V(H)$ and a partition \{$\xi_1$, $\xi_2$\} of $\xi$. We establish conditions on the directed cycles and the directed paths of the digraph $D$, with respect to the partition \{$\xi_1$, $\xi_2$\}. In particular we pay attention to some subestructures produced by the partitions $\xi$ and \{$\xi_1$, $\xi_2$\}, namely $(\xi_{1}, \xi, \xi_{2})$-$H$-subdivisions of $\overrightarrow{C_{3}}$ and $(\xi_{1}, \xi, \xi_{2})$-$H$-subdivisions of $\overrightarrow{P_{3}}$. We give some examples which show that each hypothesis in the main result is tight.